You're going to need a camera with a couple things:
1) High frames per second (fps) capture rate. At least 5 to have any chance of capturing key frames.
2) Long focal length to fill the frame with players. 200mm minimum, more depending on the field/sport.
3) Good high ISO performance - you're going to be trying to shoot action photos in some of the worst lighting you've ever seen: gyms (for basketball, volleyball, wrestling, etc.), outdoor field lighting, etc. You have to shoot with high shutter speed to stop motion blur from ruining the photo, yet you're (this is pretty much every hobby photog, not just you) not rich enough for the fastest glass, so up the ISO goes.
I'm a Canon shooter, so I'm going to, naturally, recommend a Canon camera. I'd pick up a T4i kit with the 18-55mm and purchase the 55-250mm EF-S lens with it. That'll break your budget, however. B&H (my preferred vendor): »www.bhphotovideo.com/c/product/8 ··· T4i.html
for the T4i kit, 55-250: »www.bhphotovideo.com/c/product/5 ··· _IS.html
(check local stores, you could potentially find it cheaper, but it might also be the "old" one)
Until you start getting into it really, really big, you're not going to notice much (any) difference between any entry level cameras from the different companies. I know Canon, so I recommend Canon. The Nikon offering is probably just as good and so is the Pentax.--
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