Any of the following cameras will allow for sufficient low light focus capability and high ISO performance to allow capture of something like indoor sports, and fit in your budget:
Pentax KX
Pentax KR
Canon T2
Sony A550/580
Sony A33/55
Nikon D3100
Nikon D90
Nikon D5000
However, ALL of those cameras will be fairly dependent on pairing them with the right lens - a lens that has to be reasonably fast (ie: a large maximum aperture) and have the reach needed for where you'll be shooting from (only you can determine that). The lens, especially as you get ones with fast apertures, will easily cost you more than the camera body, and be large and heavy the faster they get. How fast you are willing to get depends on what you're willing to spend and carry.
You could probably get pretty reasonable, usable results with a somewhat basic zoom that goes up to around 100mm at F2.8...those can be found for more reasonable prices within your budget range - how inexpensively you can get the camera body will determine how much room you have left for lenses (the cameras I listed above will range from around $450 to right at $1000). You'd likely do better if you could get an F2.8 zoom that can go to 200mm or so, as it will allow closer detail from farther away. If you can sit fairly close, you could save some money by looking for a prime lens (no zoom - just one focal length) of 50mm to 90mm or so, and those can be pretty fast apertures (F1.4 to F2.8) for as low as $50 on the used side to over $700.
Either way, even with a basic kit lens that comes with the camera body for $500 or so, you'll still get MUCH better capabilities than a P&S camera will give you - DSLRs are much faster to focus, faster to shoot, and can shoot at ISOs up to 3200 in low light conditions with decent results. The ones I listed above could even be pushed up to ISO6400 and still produce usable results.
But your own technique and skill will be very important factors - learn more about photography and how to control your camera, and you'll get much better results from any of these DSLRs than if you just stick it on Auto and press the button.