Oh, I forgot one more thing about the settings...
Don't forget to set the white balance as indoor gyms have terrible lighting with their funky lamp bulbs (fluorescent, halogen, etc.). Usually auto white-balance works well if you use a d-SLR camera. Take a couple then check the LCD. If the colors don't look too off, it's good enough to start out and it's tweakable with software. Otherwise, keep changing the WB modes around to achieve something you like.
BTW, d-SLR and good fixed lens might require a learning curve but if one doesn't do something drastically wrong (or just use full auto) to start out, the results will be better than (if not superior to) any point and shoot at a meet setting. But, what a good camera gets you (good fixed lens will also do) is to ability to tinker with ease if you choose to. (At which time, it would turn into a hobby) Since your motivation is to yield good results at a meet, P&S will never get you there and you will never be happy (I have a P&S as a backup so I'm not entirely wet). As one who isn't a natural shutterbug and has no prior training, it took me a lot of trial and error just to get to where I am, which is far from professional. Needless to say, I still have much to learn.
After that being said, I agree hitting a good camera store is a good approach. But, for most small town USA, that is pretty much a once-upon-a-time thing. Even Ritz closed most of its stores around the country.