(sorry that I haven't posted in a while, life's been rather hectic)
In my opinion, before confronting your family to tell them that you would like to become vegan, it is ESSENTIAL that you really know what you're doing. Read up not only on the accounts of agribusiness' cruelty towards non-human animals, but also on PLENTY of nutrition-related reports and articles.
Actually, since many parents oppose veganism mostly because they think you won't get enough nutrients ("Where do you get your protein?"), it's cruicial to learn everything you can about the health benefits of veganism. Make sure to check articles about each specific nutrient that you know they'll ask about, like B12 and iron and calcium. Then, when they ask you these questions, you can respond intelligently and be taken much more seriously.
This doesn't just apply to teen vegans, though- any aspiring vegan can take this advice.
When you say "I'm vegan because I don't believe in the exploitation of non-human animals, because I believe that it is the way humans were meant to live, and because there's no nutrients you can't get from not eating animal products", you'll probably sound much more well-informed and be taken much more seriously than if you say "I'm vegan because I don't want to eat animals", even though both of these explanations make perfect sense.
Here are some *incomplete* lists of books and sites to check out:
Diet for a New America by John Robbins
Becoming Vegan by Brenda Davis
The China Study by T. Colin Campbell
Vegan Freak: Being Vegan in a Non-Vegan World by Bob and Jenna Torres