My priority would be finding a nice reef in good shape with a lot of life. If it has sharks or big fish, that's a bonus, but not a requirement.
If you want a live-aboard, and to see reviews of a couple of popular options:
Sun Dancer 2 (Belize) May 2015 -
http://www.scubaboard.com/forums/li...ving-belize-via-sun-dancer-2-may-2nd-9th.html It's said Belize diving is better out away from the mainland, in the Turneffe and Lighthouse atoll regions. You've got land-based options out that way if you wish. Some people have an interest in mainland attractions, like a rainforest walk and visiting Mayan ruins; I've seen people talk about Placencia and Splash Divers - no personal experience, but on the basis of good report that's who I'd look to 1st if that were my interest.
Cayman Aggressor IV May 2016 -
http://www.scubaboard.com/community/threads/cayman-aggressor-iv-trip-may-2016.527640/
Picking a Caribbean Live-aboard -
http://www.scubaboard.com/forums/li...ats/495738-picking-caribbean-live-aboard.html
Pay attention to the discussion about Turks & Caicos, particularly as regards to the deeper diving that may entail vs. other options.
There's now a Roatan Aggressor, and they been offering a sale; don't know if that's still going. Roatan has a rep. for nice reefs, but not so much fish life from what I read; haven't been. If I were going, I'd consider CocoView (south coast, nice shore diving, they don't allow little kids), or Turquoise Bay Resort (budget, lots of diving, obviously popular with some local trip organizers in my area, gets good review). Anthony's Key Resort is well-spoken of but seems pricey to me, unless you take advantage of one of their special that involve 2 people - I'd probably be solo.
If you do the Bahamas, one option is the Bahamas Aggressor, which does Grand Bahama/Tiger Beach part of the year. A lot of divers and cheaper than the AquaCat; not sure what you're view is on diving with tiger sharks.
For land-based, I was impressed with Key Largo,
and got in 4 dives/day with Rainbow Reef Dive Center. Their location moved a bit since then, but it should still be one of the good options.
Richard.