Carnival Cruise to Grand Cayman, Cozumel, Mahogany Bay/Roatan, Belize — best ports for excursions?

Husband and I are planning our first cruise and trying to decide which ports are best for excursions and which are best to get off the boat and explore on our own. We had friends who recently took a similar route, and highly recommended a specific ruins tour in Belize, and also said Cozumel was really dirty and was the only place that they felt unsafe not being on a cruiseline-endorsed excursion (that’s coming from a cop).

We’d like to see ruins and go snorkeling. As for exploring, we’d love to wander through a town/city on foot to get lunch, see the local scene, possibly grab a drink, photograph the scenery. We’ve done a lot of work in developing nations and are not into the exploitation of nationals and photographing them—just ourselves, scenery, cool backdrops, etc. We want to know your favorite restaurants, bars, hangouts, etc. Any and all advice/tips/tricks for these 4 ports is greatly appreciated!

ETA: How is the drinking water quality on most cruise ships? Is it tolerable, or does it taste like bad, desalinated water? Better to buy bottled water on board?