One of the first purchases I made after becoming Open Water certified was a point and shoot camera (Canon PowerShot SD1100IS) and an underwater housing. I enjoy taking pictures in "real life" and wanted that to carry over into diving. A lot of my pictures really sucked, though, and I wanted to get better.
Unfortunately, I found a lot of the information on the web is for expensive SLR cameras, with tips pertaining to things like aperture settings and strobe placement---not things relevant to my little cheapo. As a result it's taken me a while to figure out some of the basic do's and don't. Here's some of the things I've picked up that I hope save you some time.
Under water is the same as above water, mostly
Things pertaining to framing, composition, lighting, and so on apply equally well underwater. Particularly that you should be taking LOTS of pictures. Memory is cheap, so load up.
Learn your camera
Even my simple camera has tons of features that I could easily never find. Go to your camera's manufacturer's website and download the manual. Browse the table of contents for anything of interest. Keep an eye out for things like "scenes" or "modes". For instance, my camera has an underwater setting that compensates for the strong blue tint everything has. Bingo!
Don't use digital zoom
It looks like crap above water, and it looks like crap under water. I know that stingray is swimming away too fast to catch, but digital zoom isn't going to do much to capture the moment.
Learn to use the macro mode
The manual should tell you how to activate macro mode, and also when it should be used. For my camera, that's when the object of interest is between 1.2in and 12in away from the lens. For everything else, use the normal, or "infinity", mode.
Use the flash when appropriate
When you are taking a picture of something less than about 3ft / 1m away, use the flash to put some color in your photo. Your underwater housing probably came with a flash diffuser, make sure that's connected.
Don't use the flash when inappropriate
If your fish is farther than about 3ft / 1m away, don't bother with the flash. It will only cause backscatter, which is where light reflects off all the floating particles in the water. Even water with 100ft visibility will have it.
Get fast with your camera
Part of learning your camera is getting fast with it. You will most likely not keep your camera in macro mode the whole dive, nor want the flash off the entire time. Learn to turn these features on and off quickly.
At the beginning or before the dive, figure out what you want your "home" settings to be: those that the majority of your pictures will be taken under. This can be based on the environment of your dive or just past experience with the pictures you usually take. From my experience I can tell you this should probably be flash on with camera in macro mode, but it can vary with the type of dive.
Be liberal with video
Video mode on point and shoot cameras is a godsend for diving, and it doesn't take up nearly as much memory as you might imagine. It only uses about 1MB / 1 second, so keep rolling! Practice keeping a steady hand, cause trust me, your exciting footage will look terrible later on your computer where shakiness becomes painful to watch.
Adjust white balance
Almost all cameras have a manual white balance, where you point at a white object and click the shutter.
When underwater, everything takes a blue tint, so carry around something like a white dive slate. When you reach depth adjust the white balance by shooting at the dive slate. All pictures from then all will automatically be color corrected!
These are just a handful of tips for getting started. As I learn more, I will share what I learn here, including more technical details, possibly using pictures I take as case studies.