Our very first cruise was on the Carnival Destiny in 1998 and we did the same route. This is what we did:
Jamaica - We walked off the ship and hired a gentleman by the name of Errol Issacs...he was wonderful. He drove us around Ocho Rios, to Shaw Gardens, Dunn River Falls - we only walked down to the water to see the falls, we did not climb it though...then Mr. Issacs took us to lunch at a place called Bibby Bips (it was in a cinder block building and one of the finest meals we have ever had in our lives!) and shopping. He was so nice and kind and at no time were we afraid.
Grand Cayman - We took a cab to the Westin Hotel and hung out at the beach, then went back into town and had a very expensive lunch of turtle and conch - all good. Seven cruises later, the water in GC is by far our favorite to this day. I'm not sure if the Westin still lets cruise ship passengers go through their hotel...the beaches are all public...you'll just need to find another access...I think.
Cozumel - We are going there for our 5th time since 1998 this coming Feb. I love Cozumel. The only real excursion we've done though is the Tulum Mayan Ruins and that was in 1998. We were dropped off at Playa Del Carmen at around 7 AM on a tender ship (our ship continued on to Cozumel - which was a little scary but we survived!) after reaching the pier and disembarking the tender ship we walked up about two blocks and got on a bus. We then drove for approx. 45 mins to Tulum. Take your swimming suit/trunks because once you reach Tulum and after you tour it you can walk down a hill and swim in the beautiful water. We then returned back to the ship, it was around 2 PM at this point, had room service deliver lunch, took showers and headed to the downtown area of Cozumel. We then toured a museum downtown that is in a beautiful building and you'll learn even more about the Mayans. It was $3 US. We then ate at Palermas Restaurant....and then headed back to the ship.
Since then we've always gone to Palermas, and just walked from ship to downtown, we go to the grocery store to buy vanilla and we've gone to Dzul Ha beach which we love because it is sort of deserted.
My s-i-l did a glass bottom boat in Cozumel and said that was very, very nice.
I hope you enjoy your first cruise and if you have any other questions...ask away.
