St. Lucia by the Pitons - in the little bay in front of Anse Chastanet, is terrific. Unlimited visibility, calm waters, and a wreck as well.
I have also gone at Grand Cayman and Martinique; neither was as spectacular as St. Lucia in January. In Martinique, the visibility in March was so-so - probably about 40-50 feet, and it did not seem as colorful as St. Lucia. Grand Cayman in Februaryhad calm water and unlimited visibility, but we did not see the sea life we'd hoped -note, however, we did not do a wall dive.
Also, while not technically "Caribbean," the Yucatan Peninsula in Mexico down to Belize is home to the 2nd longest barrier reef in the world. Another dive worth a try is a cenote dive in the Yucatan. They are sort of like cave dives for beginners - truly fascinating. Cenotes are not technically "caves" because you are (technically) able to see light shafts (and hence (technically)have access to air) at all parts of your dive. That being said - it is still rather challenging and nothing like open water diving, and you should definitely go with an experienced guide.
@Marilee, that is a tough one. They did sink a few ships off the coast of Miami, but the real problem for me is water temperature.
You could try diving outside of Tampa with the manatees. A manatee, though one of the nicest creatures around, is kind of scary, when you are hanging out in the water with him/her, your tank, and your dive buddy. These dives would be more shallow (35 feet). But it certainly is different. I don't really have a suggestion for you on a company as we'd just go out on the boat when I was a kid and do it, but I'm sure you could google and find a great place. Plus Central Florida's cheap and people are really nice.
Having dove the Caribbean and its surrounding waters for 43 years (Of course other sites as well). I can give you my favorite sites for good diving throughout all the waters big fish are more and more hard to find because of overfishing.
Cayman Brac (closer to the wall) or Little Cayman
Islands off Belize
British Virgin Islands
Turks Islands especially Pine Caye
Addition to the above Anse Chastenet has a very small beach which is not private and I found overcrowding to be annoyiong.
Saba has great diving because it is in a fish preserve but diving time is not long beause you are mainly diving pitons that start at a depth of 80'. It is the top of a volcano peak 3000' above the water with little to do other than dive and read. Restaurants are limited secondary to the small size.
Just to expand on Hippidoc's post. Belize is the 2nd largest barrier reef in the world (behind Australia.) The diving and snorkelling are amazing. Anegada in the British Virgin Islands is supposedly the 4th largest reef. (We were just told this while visiting there in September.) Anegada's reef is basically off shore so you really can snorkel right off the beach. For Belize, the best option is a live aboard such as The Aggressor. They will take you out to spots such as the famous "Blue Hole."