IMHO the key to getting decent shots underwater is to have a camera that is easy to shoot in manual, and to have at least one strobe. This is so that you can select a fast shutter speed to eliminate ambient light and light your subject with a strobe.
I totally disagree on the "manual" requirement. It all depends on what type of shots you want. I really don't like to cut out all ambient light by shooting at small apertures and fast shutter speeds, because I shoot wide angle and want the ambience of the natural light in my pictures. If you shoot macro and want to isolate your subject against a black background, it's a totally different story, and manual is probably a good way to shoot.
Go Pros may be nice toys above water but you can't attach a strobe to them. (If you try to shoot in automatic mode, your camera will meter for ambient light and choose a slow shutter and wide open aperture and you will get no-where fast).
Again, it depends on where you want to go. My primary beef with GoPros for stills (and video, too, BTW) is that you can't get raw format, and the optics are pretty crappy. Micro-sensor, JPEG only and fix-focus lens? Makes snapshots easy, but isn't good if you want to get the most out of your shots. I've got one myself, though, to be able get some snapshot-/vacation pictures-quality video footage for myself and my family.
My suggestion is get a Canon G16 or S120. Or if budget is tight, you can get an older one (G15, S110) as long as you can find a housing for it. The image quality will be practically the same.
This, however, is advice I agree with. But make sure to buy something that shoots raw format and learn to post-process the raw files. If you're serious about getting shots that "pop", nailing exposure and color balance on JPEGs is difficult. You get a lot more latitude by shooting raw, and color-balancing in post is a lot easier than trying to nail it in the water.
Henrik, I see you don't have too many dives in your log yet. My advice is to go simple, as simple as you can. Contrary to common (and sensible) advice, I started taking pictures straight after my OW cert. No, I don't recommend that. However, if you're comfortable in the water and has decent buoyancy control you can start shooting pretty early as long as you use a simple point and shoot (like the Canons mentioned by others in the thread) and stay aware that your primary concern is to dive safely and keep track of your buddy. I started with a Canon G9 in a Canon polycarbonate housing, without any strobe, and you can in fact get decent pictures that way if you choose your subjects well. You can also find out whether you want to do macro (advice: get a patient buddy!) or WA. I shot program or aperture priority auto to reduce my task loading and was very diligent that my shooting didn't interfere with my diving. I allocated only any spare mental bandwidth to the shooting, having the camera tethered to my BCD so I could just drop it and forget it if anything came up. It was my starting point, and it was a great learning experience. And from the start I became really motivated about working continuously on improving my buoyancy and trim, since good buoyancy and trim are essential for being able to get the shots you want. Another good thing was that if I found out that UW photography wasn't for me, I hadn't wasted a lot of money.
After that, it kind of snowballed, and I went all the way, first with one simple strobe, then a system camera setup (since the compact couldn't deliver decent IQ with the ISO speeds I needed in dark Nordic waters), then one, and then two powerful strobes, and different lenses and ports. But I don't regret starting simple, and I don't regret spending my money on different lenses and setups, because it has been a learning experience. And I still prefer shooting WA with some ambient light in the background. I still shoot ambient only on occasions, even if I've got two pretty hefty strobes on my rig, and I never bother shooting manual. Aperture auto with some underexposure dialed in for the ambient part of the lighting works just fine for me.
Here are some of my early shots, taken with the G9 compact and no strobe (your buddy's diving light can be pretty helpful sometimes):
m43 rig, rectilinear WA, ambient light:
m43 rig, rectilinear WA, single strobe:
m43 rig, rectilinear WA,two strobes:
m43 rig, fisheye,two strobes (I'm still working to master the special challenges of shooting FE with strobes):
