Alright - couple of issues here.
1. Even if they're not going to be actively participating with you, you're going to have to have your parents on board for the idea. They can purchase your gun for you as a gift but you're not going to be able to go out and purchase yourself.
2. Knowing your income level is important. I know I had a job at 16 but it was part time and nearly 15 years ago so I was only making $50-75 per week. I'm sure with more aggressive hours and adjusted slightly for inflation you could do a bit better than that, but it really depends. Getting a gun on that budget would be hard (considering most 16 year olds have at least some other things they want to do with that money), but not impossible. If you could convince your parents to get your initial gun as a birthday gift (or even Christmas, though that's a ways off) then that would help.
Now, which sport you want to shoot plays a big factor in gun choice. If you plan on paying for all this yourself, then I'm going to suggest a .22LR and Steel Challenge. Its fun and fast, and a great way to build skills. A used Ruger Mk2 can be found for $200-250. If you want you can even get a Ruger 22/45 new in that range. In the rimfire (.22LR) division you won't need a holster nor mag pouches. Just the gun and 5 mags or so will have you ready to rock. If you want you can also get by with promotional ammo for only a couple dollars per box. of 50. That would be the most economical.
Now, the other options: IDPA or USPSA, will cost you a bit more. Centerfire (9mm minimum) is a requirement, which means that ammo will cost more - even if you handload, which I'm reluctant to suggest as an activity to a teen without a mentor. You also typically have a higher upfront gun cost, and more gear to buy (holster, mag pouches, etc). Production (USPSA) or SSP(IDPA) would be where you want to compete, but overall even there both these sports will cost more to shoot than the .22LR in Steel Challenge. Not that there's not a lot of teens shooting successfully in these sports, it's just that most of them have financial backing of a relative to support them in competition.
Whatever you choose - good luck, and don't feel hesitant to come back and ask more as questions arise.