I just recently decided to go vegan. My first real full vegan day was just last Saturday, but I had been working towards it for awhile before that. First I stopped eating meat, than I started eating less dairy and eggs, and then my parents figured it out so I decided to start the real thing.
For me, the hardest part so far was telling my parents. I was really nervous about it, even though I knew they wouldn't really have a problem with it. I started mentioning veganism (I talked about how my guinea pigs were vegan, and look how happy they are!) and dropping hints about it, and they eventually caught on and my mom asked whether I was trying to be vegetarian. She was pretty skeptical, but she accepted it when she realized I had done research and knew how to get all the nutrients I needed.
Definitely do research first! Your family is bound to ask some questions, like "Where will you get your protein?" and "What about calcium?" and all that. If you're prepared with intelligent answers to these questions, they will be much more likely to accept it without worrying about you.
It can be really hard to look at meat, and smell it cooking, and know you won't be eating it. I totally know the feeling, I really love eating meat and dairy, but you just have to remember the reasons why you made the decision in the first place. Instead of thinking of that yummy steak, think of that poor cow.
Sorry this response is way too long! I hope I helped at least a little bit. Good luck becoming vegetarian, and congratulations on making that choice!