It is difficult to explain why murder is wrong if eating meat and wearing leather isn't wrong. The most obvious answer is that non-human animals are not capable of suffering (including physical pain), like Descartes claimed. However, modern science makes it clear that for birds and mammals (except possibly those with severely devolved nervous systems, such as naked mole rats), and probably even for octopuses, that's not the case. And I think almost everybody would agree that, even in Descartes'es time, there was no reason to think what Descartes claimed. Furthermore, if the way products are produced today affects at all the ethics of buying those products, then it is unethical to buy eggs from factory farms. Most of the antibiotics today are used in the egg industry (it's hard to tell the exact percentage, but it is probably around 70%), and that causes superbacteria. If you want to do the number one thing that will reduce the chances of a pandemic of superbacteria occurring within our lifetimes, either stop eating eggs or find some small farmer which you are certain doesn't pump his chickens with antibiotics and buy eggs from him.
What's the best way of spreading this message? I wish I knew the answer. However, I have years of experience of spreading that message, so I know what not to do, at least not to be dangerous. So, here are those things:

1. Don't yell unrelated fringe claims, such as Flat-Earthism, radical anarchism, global warming denialism, denying massacres, anti-vaxxer stuff, and so on.
   As for Flat-Earthism, look, if you so blatantly reject the science of the shape of the Earth, what's your justification for accepting the much softer science that vegan diet is healthy? Furthermore, while there are some Flat-Earther's arguments that make some degree of sense (the one with sunrays not seeming parallel...), for most of the Flat-Earther's arguments, you need to reject logic to accept them as valid arguments, and not only science. And there is no point in debating with somebody who rejects logic. If you believe in Flat-Earthism, make sure you've watched this video I made (if you cannot open it, try downloading this MP4 and opening it in VLC or something like that). Look, I understand it, we are living in a schizoid society and most of the people are holding plenty of irrational beliefs. But Earth being round is not one of those irrational beliefs.
   As for radical anarchism... Aren't you afraid you will make exactly the causes you are fighting for worse? Who do you think is doing more to fight superbacteria: is it the few people (vegans...) who boycott the egg industry, or is it the governments who are regulating antibiotics? The answer seems obvious to me: governments are doing more. And just like the science suggests that you shouldn't buy eggs from factory farms, it also suggests (in fact, even more so) that the government should regulate the use of antibiotics. Furthermore, you are using the Internet to spread your message, right? What do you think would happen to the Internet if there were no government regulations (not even the basic regulations such as that ISPs should set up their DNS servers to respond only to the requests from the IP addresses they are supposed to respond to)? Well, I have a Bachelor degree in computer engineering, and I think I can safely tell you that, without the government regulation, the Internet would be paralyzed by denial-of-service attacks. If you believe in radical anarchism, make sure you've watched this video I made (MP4). Look, I understand it, the vast majority of regulations are nonsensible. We would be better off if the government were more restrained in how much regulation it can pass. It's unfortunate that we are living in a society where nobody knows first-hand what is legal and what isn't. And what governments around the world are doing in an attempt to curb violent crime is wildly inappropriate, and so is what the governments around the world did in an attempt to curb the spread of COVID-19. But some regulations are necessary.
   As for global warming denialism... Look up Lysenkoism. That's what happens when pseudosciences similar to global warming denialism (which try to conquer the nature we don't understand instead of trying to minimize our impact on nature) get institutionalized. Advocating such pseudosciences is incredibly irresponsible. Only slightly less irresponsible than saying that we should continue massively using antibiotics in the egg industry. (I say "slightly less" because the dangers of global warming are indeed hypothetical: we don't know what will happen in the 21st century society if the Earth gets two degrees warmer. The dangers of a pandemic of superbacteria are not at all hypothetical: we have seen what happens in the 21st century society when a pandemic occurs.) Furthermore, if you so blatantly reject the scientific consensus of the anthropogenic global warming, what is your justification for accepting the consensus that vegan diet can be healthy? Just like with Flat-Earthism, as much as you swing the pendulum to the left, it will swing to the right. Look, I understand it, hysterically banning fossil fuels without providing alternatives would lead to a catastrophy. But that doesn't mean we should do nothing about global warming. As for the anti-nuclear movement, it's a different rhetoric, but the practical effects are similar.
   As for denying massacres... If you do that, why don't you deny the existence and the extent of factory farming as well? That seems like a logical consequence of denying massacres. That's why denying massacres is incompatible with veganism.
   As for anti-vaxxer stuff... One of the best refutations of anti-vaxxer stuff I've heard is the Sunetra Gupta's saying: "Nature does vaccination all the time. We inhale dead viruses that trigger our immune systems all the time. Vaccines let us use that process for good, to make us immune from viruses that are actually dangerous.". And I hope I don't have to explain to you how anti-vaxxer stuff is incompatible with veganism. It's hard to tell whether anti-vaxxers or egg industry apologists (who claim that egg industry doesn't cause superbacteria) are more harmful to public health. I understand that most vaccines have been tested on animals, and I understand objecting to that. But harming the public health even more by discouraging people from getting vaccinated does not solve the problem.
2. Don't be anti-supplements.
   Look, I realize the supplements are unnatural and are sometimes harmful. But you need to understand that deficiencies (which can be easily cured using supplements) are harmful and sometimes even deadly. The COVID-19 pandemic was made significantly worse by widespread Vitamin D deficiency. We can politicize with the numbers, but the fact is that many people who died from COVID-19 would still be alive today if they weren't deficient in Vitamin D. It's difficult to imagine that people widely taking Vitamin D supplements would have killed so many people with overdoses. And in 1920s Valpovo, there was a study suggesting that Vitamin D deficiency was one of the best predictors of who will die of tuberculosis.
   And you need to realize the supposed dangers of many supplements are either non-existant or wildly exaggerated. For example, nobody has yet died of Vitamin B12 overdose. Not a single one. As for calcium supplements supposedly causing heart attacks... Well, that appears to only be true in elderly smokers, and not in general population.
   Also, you presumably believe that plenty of heart attacks are caused by subclinical Vitamin K deficiency, right? I am not sure how based on evidence that belief is, but let's say it's true. So, what is more likely to work: telling people to eat more fruits and vegetables, or telling them to simply take Vitamin K supplements?
3. Don't be against artificial fertilizers.
   Sorry, but artificial fertilizers are necessary to feed the growing population. It's impossible to feed the entire population using only natural fertilizers. We can only feed around 2 billion people with them. Rhizobium bacteria (found in clover) simply don't fixate nitrogen at the necessary rate to produce enough protein. And you need to understand that the so-called natural fertilizers these days are mostly greenwashing: it is, for example, cow dung... from cows fed with grains which were grown using artificial fertilizers. I understand that some artificial fertilizers are very harmful for the environment, but, to paraphrase Ayn Rand, this is entirely a technological problem, not a political one.
4. Don't demonize protein intake.
   Look, it's true that human beings need very little protein... if it contains precisely the amino acids that human body needs (like in the human milk). The protein in real food is not like that. Much less is the protein found in plant food. For many people around the world, wheat is the main source of protein, and they suffer from lysine deficiency. If you keep demonizing protein intake (saying that people shouldn't eat soy or legumes), and lysine deficiency gets more common because of that, you are responsible.
   Not to mention the supposed reasons why people shouldn't eat a lot of protein are gibberish. For the claim that protein causes your blood to become acidic and cause the calcium in your bones to go into your kidneys... Look, studies which look for a correlation between protein intake and osteoporosis tend to find little or no effect, whereas the studies which look for a correlation between Vitamin K deficiency and osteoporosis tend to find a huge effect. As for the claim that protein causes type-2-diabetes... Which mechanism are you proposing? At least when you say "Protein demineralizes the bones and causes kidney stones.", it's obvious which mechanism you are proposing. When you say that protein causes type-2-diabetes, not even that is obvious.
   It is true that there are studies that supposedly show low-protein diets help with all kinds of health problems, however, a problem with nearly all of them is that they don't control for allergies.
5. Don't claim that methionine is the real enemy of the heart health.
   It's true that methionine raises your cholesterol levels. However, people who say that don't seem to understand the scale we are talking about. The food that's highest in methionine is sesame seeds. Do you realize how much sesame seeds you would have to eat to raise your cholesterol levels by 5%? The answer, if the studies on rodents are anything to judge by, is around 300 grams per day. Hardly anybody eats that much methionine. For comparison, a diet rich in saturated fat can easily increase your cholesterol levels by 30%.
   And the moment you start talking about sesame seeds causing heart attacks on a forum where there are people educated in nutritional science, you are probably bound to be laughed out of the debate.
6. Don't claim that following some weird diet is better for managing type-2-diabetes than insulin is.
   You need to understand that, before insulin was invented, if you were diagnozed with type-2-diabetes, you only had months to live. And people were trying to manage their type-2-diabetes with all kinds of diets. None of them worked, at least not nearly as well as insulin works. Your weird diet idea has, in all likelihood, been tried, and it didn't work.
   And I think the reason why such claims (that some diet is better at managing type-2-diabetes than insulin is) appeal to many people is the same reason why anti-vaxxer claims are appealing: the time when type-2-diabetes was a death sentence happened just long enough ago that nobody remembers it. Just like the claim that unvaccinated children are healthier: it seems appealing because the time before vaccines happened just long enough ago that nobody remembers it. And another reason why the idea that low-carbohydrate diets help against type-2-diabetes appeals to many people is because people are ignorant of basic cybernetics and think that, if you mess up the controller, the control system will be fine so long as there is no noise.
7. Don't claim that some weird diet is going to cure you of cancer.
   Look, the notion that heme iron (found chiefly in red meat) mixed with omega-6-acids in the presence of the enzymes found in the human colon produces carcinogenous substances (sometimes causing colon cancer) is supported by solid evidence. However, the notion that you can somehow slow the cancer growth by starving yourself of carbohydrates and protein is not. The opinions of doctors seem to range from "Don't be silly." to "Maybe, probably not.". And it's a dangerous thing to try because, well, if you are starving the cancer of carbohydrates and protein, you are also starving your immune system.
8. Don't claim that some diet which doesn't cut the calories is going to help you lose weight.
   Not only would that require everything we know about nutritional science to be false, it is also hard to make compatible with thermodynamics.
9. Don't claim that sugar doesn't cause type-2-diabetes.
   This applies also to people who are denying that saturated fat causes heart disease. Honestly, does anybody really believe this nonsense? I simply don't see how anybody could. The evidence that sugar causes type-2-diabetes and that saturated fat causes heart disease is everything we know about biochemistry (saturated fat raising your LDL cholesterol, fructose in sugar causing insulin resistance...). The main argument against that appears to be "Here are some dubious statistics that sugar consumption reached its peak in the US back in 1990s (and there are more people with type-2-diabetes now than in 1990s)." and "Here are some dubious statistics that saturated fat consumption in the US reached its peak in the 1970s (and heart disease today is more common than in 1970s).". That's literally less compelling than Flat-Earthism. Flat-Earthers have arguments (one with apparently-converging sunrays) that are more compelling than those arguments. In case it is not obvious, those arguments are ignoring the fact that there are way more people now who live long enough to get those diseases.
10. Don't propagate coconuts as healthy food.
    You need to understand that the claim that HDL helps against heart disease, although commonly repeated, is not really mainstream science. And claiming that something that increases both your HDL and your LDL helps against heart disease is... unlikely at best. Why have scientists thus far found no pills that would cure heart disease by increasing your HDL cholesterol? It's not due to the lack of trying. It's just that pills which try to do that turn out not to actually decrease mortality when put into controlled experiments. Unlike pills which try to decrease mortality by lowering your LDL.
    The evidence for the claim that high-fat (ketogenic) diets help against epilepsy is rather thin, and saying that coconuts help prevent various neurological disorders is pure speculation.
11. Don't claim that eggs are worse for heart health than meat is.
    Look, there are two explanations as for why eggs are correlated with heart disease:
    1. Cholesterol you eat, via some unknown physiological mechanism, causes heart disease.
    2. People tend to eat bacon with eggs, which is high in both salt and saturated fat.
    If you think the first explanation is more likely, we would like to hear your reasoning.
    If you mean that eggs are worse than meat for public health (superbacteria), then, sure, they are. Way more antibiotics are used in the egg industry than in the meat industry.
12. Don't fight for the rights of animals that are probably not sentient.
    For example, fish. To claim that sharks feel pain, you basically need to reject entire neuroscience: sharks have no nociceptors. To claim that higher fish feel pain, while you don't need to reject entire neuroscience, you need to reject quite a few things we think we know about neuroscience: higher fish have little or no type-c neurofibres, which appear to be necessary to feel pain. Furthermore, to claim that fish feel pain, you need to invent an ad-hoc reason why fish don't act like they feel pain: a fish with a hole in its fin continues swimming normally. Understand that fighting for the rights of fish is not only not morally virtuous, it is also morally objectionable: it diverts the resources from what they should be spent on (on fighting for human rights and rights of sentient animals).
    UPDATE on 28/02/2024: I received some criticism on an Internet forum that I am misusing the word "sentient" here. That fish are supposedly sentient because they, for example, have eyes to see, and "sentient" means having senses. In my opinion, if that's what "sentient" means, then sentience is not a valid criterion for whether or not some living being has moral value. You know what John Stuart Mill, one of the early proponents of animal rights, said: "The question is not whether they can reason or think, but whether they can suffer."? Well, yeah, the question is not whether animals can sense, but whether they can suffer.
13. Don't claim that cats shouldn't be kept as pets because they would murder their owners if given a chance to.
    Here is a response I got when claiming that on an Internet forum: I also think that cats should not be kept as pets because they don't actually love their owners and would probably kill and eat their owners if given a chance to.That's pretty speculative. There are plenty of people who even keep large cats like lions and tigers who do not kill and eat their keepers when given the chance. Some do, of course, others abstain and choose to play, purr, etc. Unless you're asserting that domesticated house cats are actually less capable of human bonding than lions and tigers (which would be an absurd argument given the normal effects of domestication), it's undeniable that there are plenty of domestic cats who would not kill and eat their owners. I would have no problem believing that many WOULD do so, but making such a general statement puts upon you a burden of proof you could not possibly uphold and which goes against all available evidence.
14. Don't imply that milk only started to be unhealthy when we switched from grass-fed cows to grain-fed cows.
    Although commonly repeated, that claim is almost certainly wrong. Milk from grass-fed cows does contain omega-3 acids, however, the type of omega-3 acids that it contains, the ALA, is almost undigestible by humans (human liver can only convert around 5% of ALA to DHA). Around 90% of omega-3 acids in milk from grass-fed cows is ALA, and human beings can only use DHA.
    And you need to understand that, while DHA does raise your HDL cholesterol, the evidence that it translates to lower risk of heart attacks is very thin. A lot more thin than the evidence that saturated fat cause heart attacks.
15. Don't use the rhetoric "If everybody had to kill to eat meat, everybody would be a vegetarian.".
    I think a lot better phrasing of that same point is: "If slaughterhouses had glass walls, there would be a lot more vegetarians.". The fact is that most people seem to be fine with many things only if they are done by others on their behalf. For instance, hardly anybody would ask their rich neighbour to pay off their student loans, yet many people (if not most) are fine with proposing free colleges (when the government asks their neighbours to pay off their student loans on their behalf). And merely pointing that out is not likely to convince many people.
16. Don't claim rice milk is better for climate than cow's milk.
    I don't know that and you don't know that either. The fact is that both rice milk and cow's milk emit tons of methane during their production. Oat milk is far better than both of them.
    I realize that oat milk is more expensive than rice milk. However, if we are going to fight global warming, we need to be prepared that fighting global warming is going to cost us something. That's not to say we shouldn't do that. But we shouldn't expect there to always be a cheap solution to the problems such as that of milk emitting methane. The problem of global warming is very complicated.
17. Don't claim that meat is kept artificially inexpensive by government policies.
    While, in many countries, governments are indeed trying to make meat cheaper by government subsidies, the effectiveness of such policies is dubious at best. Look, price is determined by supply and demand, not by the labor it takes to produce something (look up "labour theory of value"). Milton Friedman famously argued government subsidies, in the long run, make all food more expensive as well as increasing the price of arable land and agricultural machinery (by increasing the demand for those things, as more people want to be farmers when there are subsidies). And, as far as I understand it, most economists today agree with him.
    You need to understand that it's not a rule that things which are more harmful to the environment are more expensive. Animal products that are arguably the most harmful for the environment are eggs. The vast majority of antibiotics used in agriculture are used in the egg industry, and superbacteria are arguably a bigger problem than global warming. Yet eggs are the cheapest animal products. And the same goes for alternative milks. Oat milk is more expensive than rice milk, even though oat milk is arguably way more ecologically acceptable, as rice emits a lot of methane which oat doesn't. I cannot explain to you why that's the case, but you don't need to know the right answer to recognize that pointing to the government subsidies in an attempt to explain that is a wrong answer.
18. Don't advocate for governments to ban meat because of pandemics.
    Please understand that it being illegal to eat bats in China didn't stop COVID-19 from jumping from bats to humans. In fact, it arguably made it more likely to happen, because the bats which were eaten weren't receiving basic veterinary care (as it was, well, illegal). Meat being illegal might decrease the chance of a pandemic of a superbacteria happening (if it would even do that, considering that way more antibiotics are used in the egg industry than in the meat industry), but it will certainly massively increase the risk of virus pandemics happening.
19. Don't imply that lab-grown meat is going to solve the problem of superbacteria.
    It won't because the vast majority of antibiotics used in agriculture are used in the egg industry, and we will not have lab-grown eggs any time soon. It's depressing, I know, but that's the reality. Not all global problems are self-solving.
20. Don't claim that red meat is to be blamed for the current diabetes epidemic.
    Sorry, this goes wildly against the mechanistic evidence. While there is some evidence that heme iron kills the insulin-producing cells in the pancreas, that should, if anything, lead to type-1-diabetes, rather than type-2-diabetes. The vast majority of people with diabetes have type-2-diabetes. Red meat is not an explanation for that.

I am sure there are more, but those are ones I consider the most important, and should give people a starting point on doing vegegelicism.
A good deal of them can be summarized as "Don't be irresponsible.". If you deny global warming or are being anti-nuclear, do you take responsibility for whatever happens in the 21st-century society when crops start failing due to global warming? If you advocate low-protein diets, do you take responsibility for what happens when a significant portion of the population has their immune systems compromised by lysine deficiency? If you say that some diet is better than insulin for managing type-2-diabetes, do you take responsibility for what happens when somebody actually tries to do that and their type-2-diabetes gets worse? If you are advocating coconuts as healthy food, do you take responsibility for what happens when somebody drinks a lot of coconut milk and gets a heart attack because of that? If you are talking about eggs and sesame seeds causing heart attacks, do you take responsibility for what happens when somebody replaces them with something much worse? Before saying something publicly, always ask yourself what could go wrong because of people taking you seriously.

UPDATE on 12/02/2024: In case somebody asks you why you are a vegetarian, I think a good response is: "Please, let's talk about something fun, not about farms where they torture animals.".

UPDATE on 23/04/2024: I have been thinking about this recently, is effective altruism compatible with veganism? I don't think it is. Consider the problem of superbacteria caused by the egg industry. Veganism suggests that we should buy some expensive alternatives to factory-farmed eggs, such as eggs from pasture-raised chickens or cooking with avocados instead of eggs. Effective altruism, as far as I understand it, suggests that we should buy the cheapest eggs possible (which will, of course, be ones from factory farms) and give what is left of our money to the scientists who are looking for new antibiotics. So, I think they are incompatible. And I think that the solution suggested by veganism is better because, well, how do we know there even are useful antibiotics left for scientists to find?