I booked Carnival Legend on Dec 6th, cruising through Cozumel, cayman and Ocho Rios. Is there any activity highly recommended in these places?

Here are things we liked:
Cozumel: Tulum
Grand Cayman: snorkeling with the sting rays
Ochos Rios: Dunns River Falls, rafting on the Martha Brae, plantation tour

If you go to Tulum make sure you take the ship's tour! Tulum is on the mainland so you take the ferry to Playa Del Carman then a long bus ride to the ruins. The ship will wait for it's own tour but not for you if you do it yourself. This is an all day tour. Or you could go to Chananaub, a national park. This is on the beach with restaurant, very good snorkling, a museum, snuba, etc., and they have swim with the dolphins which is fun to watch. This is an easy cab ride from the pier.
In Cayman you can easily do swim with the rays by yourself. Captain Marvin is great for this and an easy walk - turn left when you get off the tender. He is also a LOT less than ship's tour.
In Jamacia you can get a cab to take you to Dunn's River Falls. Take your camera- the guide will carry it- or get a throw away FUGI camera for great pictures.
If you go to cruisecritic.com, you can search for info about Capt. Marvin, Chankanaub and other tours.

Actually the ship is carnival liberty...
thanks for the tips.

In OR definitely do Dunns River Falls. We took boat excursion to the falls, which isn't far from where our ship docked, then had time to shop. In GC took a tour of the island and to turtle farm and then shopped. I like to shop!! On Cozumel took a cooking excursion thr the ship which was awesome. The chef was great and then we got to eat what we cooked on the beach with china,silver and crystal. Best excursion we have ever taken!! Also in Cozumel you might want to eat at Pancho's Backyard.