Cayman is a definite must with its walls, wrecks and reefs. If you can get on the north wall, go for it as its much more pristine. Normally though, its pretty rough out there, so chances are youll be going off seven mile beach somewhere. If you ever do a longer trip, be sure to spend some time on Little Cayman and Cayman Brac. Really awesome walls over there!
Cozumel from my experience (only a few times while stopped on a cruise) is different. Ive always ended up doing drift dives. The boat will drop you off and you basically ride along the current with your group, and the boat will pick you up when youre done. The fish seemed really huge compared to other places Ive dove. Problem is that you have to stay with your group (divemaster is your leader lugging the dive flag around) and if you want a closer look at something its almost impossible to stop thanks to the current. It can be a little overwhelming for a newbie too since you have to move fast to stay with the group.
Havent dove the other ports, but when I was at Costa Maya it was a fairly new one. Diving here would probably be similar to Cozumel withouth the current (or maybe even with it???). Belize is supposed to be good. They used to push it for the blue holes.
Grand Cayman and Cozumel are great places to dive. Since you are going to be on a MASSIVE ship and at least in Grand Cayman, you will be in on a busy day, I suggest booking your own dives and aviod the ship dives. A newbie can get lost in the shuffle on a crowded cattle boat.
In Cozumel I went with Eagle Ray Divers and was the only diver on the boat (my wife, or friends and several others were snorkeling off the same boat). They would drop the snorklers off and then go out deeper and drop the divemaster and myself off. Well run trip. www.eagleraydivers.com
In Grand Cayman I took the ship dive, there were only supposed to be 6 of us and the crew, but because of high seas on the North end of the island, the diver boat to there goat cancled and they put 20 more people on. No injuries or accidents, but a little tight on space to gear up. Princess uses Don Fosters on Cayman. They dive master and crew were great even with the last minute change. Would use them again. But ask at your local shop and check the internet.