Nuckelbuster has a very good reply so I only have a few thoughts to add.
You may not convince your family to become nudist. The best you can do is present the best information you can but ultimately the decision is for the individual to make based on their comfort level. Hopefully your family is open minded enough that you're even able to talk about something as potentially taboo as nudity in the first place. You'll definitely want to educate yourself all about the naturist/nudist lifestyle, as well as thoroughly write out your thoughts and feelings on the matter so you can say why living clothes-free is important to you. A good place to start for information is to do a search for "205 arguments and observations in support of naturism", a long document by The Naturist Society.
In the end, you could consider yourself very lucky if your family was open to the idea of just you going nude, making it an individual choice in the household even if they don't go nude themselves. If that should happen, accept that policy happily and be the best nudist you can. In time perhaps they would open up enough to try it for themselves, but if not at least they would learn by seeing you nude that nudity can be perfectly innocent, comfortable, and clean living. You'd be amazingly lucky if you convinced them to all live as nudists, but there again it's a personal choice to change one's lifestyle. And if they don't want home nudism at all, you'll have to be content to keeping your nudity in private places and times.
I started into nudism when I was about your same age but my family was terribly close minded and immature about nudity. I tried talking to them only once about it but it didn't go well at all. Hopefully your family is more willing to listen and discuss things rationally.