Bear in mind that the children will not be allowed in the pools on the ship unless they are potty trained.
Excursions...Grand Cayman is a tendering port which means you will be taking a smaller boat from the ship to shore. Once there I recommend the Turtle Farm. For Cozumel I'd go to Chankanaab State Park. You can google these and get more details.
There isn't much I can suggest for Jamaica. At Labadee you are on Royal Caribbean property the whole time so once again it is primarily a beach day.