Let me start by saying that idea of convincing someone is already not a good start. Needless to explain that once you try to prove someone wrong that person won't hear a word you gotta say. I've been an animal rights advocate for the past 2 years and what I've noticed is that I make the most difference when stop trying so hard.
One of the biggest problems is that most of the people are very misinformed about meat&dairy industry in terms of it nutritional values, what it does to the environment and what animals go through to get on our plates. So one thing I always say is that I can offer to share the information I have. For the past 6 years I have been doing plenty of research, reading study after study and watching endless documentaries to make sure I am educated and can help other people if they have ANY questions. If you eating animals it's only fair to hear their side of the story, or if you care about your body and health there are lots of information that those who in power don't want you to know. Therefore being educated about veganism is a very first basic step of making YOUR conscious choice. Not because thats what you've been told to do by your parents/school/doctors/government but because you know enough to choice what works for you and if a taste in the mouth worth someones life of suffering.
Another way to promote plant based diet is to show friends&family how easy and delicious our food could be. Veganism is not about limiting what you eat, because any food can be done in vegan version. Market is growing so fast that there is replacements for any kind of cheese or meat you will ever want. So I would cook for my friends, take food with me on planes, order vegan food at house parties, ask about vegan options at restaurants and cafes.
Personal approach is really important too. What I do is analyse whoever I am talking to. Some people are so used to their own ways that in the very beginning of debate I can sense it's not worth my time. Other people are curios and for those I will be giving all I have. If it's a person is into sport I will tell them about all the world best athletes and champions who are vegan and about how much better it is. If it's a mother I will tell that dairy industry steals cows babies to get more milk. For young fashion loving girls I will be advising all the brands that don't test on animals. For dog or cat lovers I will try to explain that the way animals feel emotions is exactly the same with pigs, cows or domesticated animals.
Social network is a very powerful resource but slightly tricky because since people tend to react to positive messages it's a hard job to talk about animal abuse and violence and suffering without pushing people away.
I can go on for many hours about it but this is the main things that came to my mind after 2nd glass of wine : )
Even though the question focuses on diet, I would de-emphasise that initially and start with vegan philosophy - the thing that drives a vegan's diet. Veganism is a radical justice-for-all idea. The vegan social movement originated in England in the 1940s when WWII was still raging. The early vegans saw veganism as part of the peace movement and integral to the moral evolution of humanity.
Vegans are pro-justice - for all sentient beings. Just as Professor Tom Regan is credited with getting the notion of rights "over the species barrier," vegans are trying to do exactly the same with the notion of justice. If we believe that other animals are morally significant - and vegans would regard them as moral persons - then we cannot use them in the many ways in which we do - animal use represents rights violations.
The 21st century are exciting times for vegans for, although some animal advocates want to limit the meaning of veganism, there has been a most welcome emergence of intersectional veganism in recent years. VegfestUK held a brilliant pro-intersectionality conference in London in 2016 which saw many people of colour finding their voices within the vegan community perhaps for the first time. These more political vegan activists reject the stale mainstreamness, and the racism, sexism, and ableism of the likes of PeTA and embrace grassroots community-based advocacy models which involve making links with other social justice movements.
So, if we want people to reject their diets based on flesh, dairy, eggs, and other animal products, I say that we show them the radical vision of the future that veganism provides. A vegan world would be radically different to the one we have today: justice not profit will be its driving force. Veganism is pro-feminist, anti-racist, critical of ableism in, for example, language and body shaming, and a vegan world would mean that every human animal on the planet would be fed and have their rights and dignity respected. As for what vegans eat, it's justice-for-all stance means that we do not violate the rights of other animals by eating their bodies - or by using them in any other way.
The co-founder of the Vegan Society in 1944, Donald Watson, suggested that veganism is the greatest cause on earth. Fully understood for its radicalism and intersectional thrust, I believe that to be true.