People get very nervous about underwater photography. Not so much about the actual act of creating the photograph, but about getting it right. I am here to tell you to stop worrying!
You can have the most expensive photography equipment and every doo-dad — and goodness knows that I travel with a lot of gear — but you don't have to have all that to create really nice photographs for your own personal memories.
The most important thing to remember is to breathe. Literally, breathe.
All too often I'll watch newbie photographers trying to get the right shot and they will forget that they are underwater. You have to remember that you need to breathe properly. That means more than just taking in and out air, it means staying calm. Take your time compose the shot and take the picture.
There are a few other things that are as simple and that you should understand. For example, how light moves through the water column. The deeper you go, the more you lose warm tones. That means you start losing reds, oranges, and yellows as you go deeper. The easiest way to correct this is put an appropriate filter over your lens. If you are in the Caribbean where the water is a little bit bluer, you're going to want to use an orange lens to balance the color. If you're in some greener waters, you want to use a red filter. An easy way to remember this is that red is a complement to green; orange is a complement to blue.
If you're just starting out, keep it simple. I know many people who use GoPros and they can be really fantastic for video, but if you're looking to get still shots go with a point-and-shoot. My big rig with all my strobes and housing — known as "Baby" — has a little sister named "Baby Blue". She's a little Nikon Coolpix that I keep tucked in the vest of my BC and I get some great still shots and as well as some wonderful video with it. This powerhouse can go to 100 feet, has an underwater setting, and is great topside as well.