My all time favorite port is Cozumel, Chankanaab park is a great place to go if it is your first cruise, as its a short taxi ride to the park and there ar e plenty of activities to do, with restaurants, snorkeling, dive huts, Dolphin Encounter, play areas not to mention a nice garden to walk through, great photos.
Grand Cayman is nice, I haven't yet found a knock it out of the park thing to do there yet but I did enjoy doing an island tour last time of all things. The turtle farm was neat too, my two girls liked that a lot.
Jamaica was interesting, it is nice but I found the locals a bit overwhelming, they are not the least bit shy of asking for tips and what not and if you wander into the shop areas then be on your guard as they are very slick in getting you to buy stuff. As a result I left Jamaica with a very bad taste in my mouth but I want to go back and try again. We did the Dunn's River Falls thing which in hindsight wasn't that great for me because I live in North Georgia and have climbed up and down waterfalls, rapids, and rivers all my life so it wasn't anything new for me. You can save money by not booking excursions through the ship but I don't think I would recommend that in Jamaica, especially for your first trip, I got a little freaked out at one point. Not trying to scare you but as with any where you go caution is warranted and in Jamaica I felt more of a need to be on my guard.