Whether you go vegan for the animals, the environment or your health (or all of the above) is totally up to you. You could also just choose to reduce your consumption of meat and dairy and that would be an awesome thing to do.
These steps are not to convince anyone (you can do that by yourself), but rather to guide anyone who leans towards a more ethical "lifestyle".
1. Go at your own pace
Whether you choose to throw all you meat and dairy out at once and quit cold turkey or to steadily reduce your consumption of animal products is totally your choice. There is absolutely no wrong way to do it and I think that we forget this important fact all too often; it's a process, not a competition.
2. Read the literature
Because we live in a society where meat is predominant (carnist), some people will totally understand your choice and share your opinion, but some will also attack you for not doing what the majority does three times a day: eating animal products. That's why it's important to know at least a little bit of literature and facts about veganism. I recommend reading Élise Désaulnier's "Défi Végane 21 Jours" and Martin Gibert's "Voir Son Steak Comme Un Animal Mort" for francophone readers and Melanie Joy's "Why We Love Dogs, Wear Cows and Eat Pigs" for anglophone readers.
3. Watch documentaries
For the same reasons as why you should be reading about veganism, you should be watching documentaries. After all, images are worth a thousand words, right ? I personally suggest "Cowspiracy" and "Forks Over Knives", both are on Netflix.
4. Learn recipes
Changing completely the way you eat can seem laborious at first, but once you have a few good and easy recipe at hand, it becomes only more interesting to experience with new and more traditional ingredients. Check out the blog Oh She Glows for recipes that are easy to do and healthy, Minimalist Baker for recipes that are under 10 ingredients and Hot For Food if you are a #fatvegan like me.
5. See a nutritionist if needed
Transitioning from a diet to another can be hard, I am not going to lie, but there are always ressources. Nutritionists like Anne-Marie Roy and Thao Bui offer help for people who want to become vegetarian or vegan. It's also a good idea to not only eat fake-meat products three times a day, but a lot of whole food, greens and fruits to get all your vitamins (but vegan desserts don't count, right ?)
6. Get inspiration from social medias
If you don't know a lot of real-life vegetarians and vegans, it may be helpful to follow a Facebook group that is dedicated to helping people in transition. I suggest La Fourchette Végé, a closed group for vegetarians and vegans alike that share delicious recipes and good addresses. You can also follow inspiring instagram accounts such as Loounie, Odilejp and Chic & végane.
7. Take your B12 !
Seriously, it's really important. Make sure your plant-based milk is fortified or take a complement. Your body will thank you later.
8. Talk about it
When telling your friends and family about your veganism, you should sit down with them and tell them the reasons of your choice. If they have questions, answer them if you can, but you should keep the conversation open so that they know they can refer to you if they want to know more about it. Also, if you have diner at someone's place and they don't know about your veganism, you should maybe tell them in advance so they don't feel bad serving you animal products; at least, they will have time to prepare a special menu for you. Or even better, call and suggest to bring your own food or to prepare something with them. I personally love to bring my squad and family to amazing restaurants, so they also discover awesome places.