I'll wait for others to chime in with the "good", so I'll throw out something "cheap and easy". I just got a Creative Vado thing that's about the size of an iPod, costs $100 US, and is meant mostly for shooting video to upload directly to YouTube (or Photobucket, or you can do what you want with the files, but if you've got Windows it'll put it on those sites right from a program on the device itself, you don't need to even install it on your computer). Mine's neon pink, but I think they also come in silver.
Video's only YouTube level quality, it'll take 1 or 2 hours depending on the setting you choose, but it's so small you can take it everywhere and therefore might make a decent backup device if you're at a meet and something goes south on your good camera. Better than trying to use a cell phone camera, anyways, especially as it does have a tripod mount. Haven't tried shooting gymnastics footage yet, though, just people randomly goofing off at work. But low quality's probably still better than no video of your child's amazing vault when the spare battery's at home...
I am not a professional photographer, so this is a great 'point and shoot' option for shooting gymnastics.
I researched specifically for cameras that took good photos in low light (all gym meets are low light, since they are indoors), and which still took decent action photos.
It also has a video camera built into it, so I can just turn the dial, and start video-ing, instead of fidgeting around in my bag for a different device, and missing out on action shots while doing so.
The camera also has a NIFTY little function that lets you capture still photos from the videos you took! This is how I handle the action shots: I just video the gym routine. Later, I can go frame by frame, and capture the photos I like.
If you look on my album on this website ("msl529's album"), you will see a couple of photos I captured from video with this camera (the L5 bars tap swing and a couple of beam photos), and you will also see the still photos I got w/ it (family pics, etc.).
Also if you look at the video section of this website, you can view my "L5 State Video" which I took entirely with this little handheld camera. I loaded & edited it on a website called 'One True Media', and then posted it onto You-Tube.
All this can be done by simply removing the memory card from the camera and putting it into your computer (you need a memory card slot in your computer).
The camera is small and easy to carry. I never leave home without it!
I may not get super-professional photos, but I love its ease of use, and do get some very nice pics w/ it.
If you wanna go a bit pricey I bought myself an Olympus Stylus 840... It's has lots of features including burst mode and shoot and select. I had fun with the firework mode too... The pictures come out nice and sharp. The video when watching it back can be paused and moves frame by frame. It showed us last week that the reason my teammate jammed her finger on a yurchenko was because she rolled her hips... you can play the videos back too but that doesn't help with skills its just fun to watch. Its cost about $300 but its way worth it.