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 : )
xxx