Welcome to the wonderful world of cruising.
I do not have any experience with Carnival cruise line, but I know a lot of crazies here are partial to it. Hopefully they will jump in and answer any questions you may have.
When we were in Grand Caymen we went to Stingray City and swam with the rays. It was the most incredible experience ever. (I am not a great swimmer and not overly fond of the water - we were standing just slightly over waist deep)
In Cozumel we were to Paradise Beach - had a great lunch on the beach - really great day.
I haven't been to Belize in many years but do remember an enjoyable trip to the Mayan Ruins. It was not my favorite port but got better once you left the area where the tenders docked
We just made our first trip to Costa Maya in May and loved it. There is a public pool right where the ship docks but we went to Pez Quadro for an all inclusive day - beach, drinks and a lobster lunch (extra charge)
I can't wait to go back there.
As I said we have a lot of information here and a lot of different opinions about different ships and ports but we all agree to disagree so that makes for a friendly cruise site. If you do not get an aswer to your question, feel free to ask again. Sometimes the boards move so fast it is easy to overlook a question.
Happy sailing and happy planning -