I recommend the following for a higher yield while still keeping things on a budget:
a) Do not use a point and shoot. While great under perfect lighting, it does not have the flexibility or the lens you need to take a decent picture indoors (especially at a gym meet). Buy a digital SLR. Pretty much any model $300 and up (new - not including lens) would work. Of course, the more you spend the better camera you can get because of the features offered. Later models come in higher resolution so you can always crop your images to your liking.
b) Get a faster lens. The standard zoom to f5.6 will not work. It's just too slow for light to get in for action shots. A big zoom with low F stop is the most flexible but it is very expensive (say $1500/2000 and far up). Instead, a prime (aka fixed) lens at say f1.8 or lower will be cost effective. Even a standard 50mm/f1.8 stock lens for around $100 can do an OK job. The challenge then is to find a spot in the stands at the right distance to shoot.
c) Open up your ISO. The higher the setting the faster the shot and the brighter the picture. Use between 800 to 1600. If set too high, you lose a lot in clarity. A picture taken at ISO3200 is very blochy and grainy when blown up. The use of an editor after the fact can help smooth/soften some of that.
d) Get a decent editing software. To start, there is no need to spend the big bucks on Photoshop. Do some research to find a good free/cheap one (ViewNX is great for Nikon). Don't worry about underexposure by say 1 exposure. If the image comes out viewable and in focus on the screen then it's probably good enough. The software will take care of the slight underexposure really well.
e) If your hands aren't steady, use a tripod or at least a monopod (or use one anyway). Or, take shots from a fixed position.
f) Lastly, take lots of shots. Digital photos are free so why not. Practice practice practice (with both the camera and the software).
Good luck and have fun!