First, DSLR is the way to go. Canon & Nikon are the two "big dogs". You should be able to get the camera and 1-2 "kit" (basic) lenses for ~<$1000. However, that won't necessarily help you with taking shots in a gym (unless it is EXTREMELY well lit).
Here are some examples... this picture was taken with a kit lens that came with my Nikon D5000...
A year later, I learned a little bit more... same camera & same lens...
This year I rented a "faster" lens. Timed it so I had the lens for about 3 days and used it for two games...
I got the camera to take pictures at my kids sporting events. I had dreams of getting the camera and taking "newspaper" quality shots. It's not just the camera though. You'll need a good lens, and learn how to use both. As you can see, I've improved, but I'm still not 100% happy with my results.
Another website you can check out:
http://forums.steves-digicams.com/All kinds of forums with suggestions, tips, and samples.