Well, for one, you're going to need to schedule a contact lens fitting with an optometrist. They'll measure the base curve of your eye and make sure that you can safely wear contacts to begin with. They'll also teach you how to wear/remove contacts, and how to care for them. Once you buy your lenses, you'll want to bring them into the optometrist to make sure that particular lens fits properly (even within the proper base curve, it may not be a perfect fit).
Convincing your parents is something I can't help you with. I wear contact lenses to correct my vision, so as long as I buy from a reputable brand and stay within my monthly budget, they don't care (mind you, I'm over 18).
I picked a brand whose products looked good (vibrant but natural), has good reviews, and is made in a base curve that is the same as my regular lenses. I need a prescription lenses, so they are a bit pricier (1780/1980/2180 yen per EYE for 3 different brands versus 2780/2980/3280 yen per PAIR for the same 3 respective brands' plano version), so I also needed a brand that offered prescription lenses. I'm not about to wander around blind for the sake of a different eye colour.
Two of the popular coloured contact lens webshops here are "Pinky Paradise" and "Alice and Rabbit". I haven't bought from either store personally, but you can probably read reviews of their services by Googling them.
Coloured contacts are safe if you handle it properly. 1-use disposables are pretty much that - stick 'em in, throw 'em away when you're done. There are also 1-month and 1-year lenses, but maybe you shouldn't start with them?
I also don't recommend starting with circle lenses, since they have lower water content than non-circle lenses.
RIP CosRevo, Love Laugh, Sweety Pop, Cocosilver.