With the digitization of cameras, plus the popularity of the compact camera, underwater photography is much easier than it used to be. Focus on a subject and shoot away. Take as many photos as you can. Mess around with all of the settings and see what works best in certain conditions and depths. A few pointers to remember. As you descend, not only does it get darker, but the colors start to fade away. Red is the first to go. Many cameras have a red filter option to add this tone back into the photo. Use it. Get close and use a flash. The rest you can tweak with practice. If you have a compact camera–i.e. point and shoot, you will never get good photos of the reef from a distance. So stop trying. Slow down and take photos of the sea whip polyps, the bluebell tunicates, the flamingo tongues, and the christmas tree worms. The most common creatures can be new all over again when behind the lens of your new underwater toy. And think of the new artwork in the house!
I highly suggest taking a class or getting some sort of instruction from a photo pro. Most dive shops and resorts will have a photo pro who will be more than happy to offer the photo specialty or a personal critique. For those more advanced and wanting to improve, you may want to sign up for a weeklong photo week at a resort or a liveaboard where a photo pro is leading the group and everyone is in photo mode. Underwater photography is a great way to add to an already amazing hobby!