Honeymoon planning advice

Hello group,

I am planning my 3 weeks honeymoon trip, starting mid-November from
Italy.
We would like to spent a few days visiting the Big Apple, and then
going south to the Caribbean Islands.
Just got the two way ticket to JFK, now we have to choose which islands
to go for the remaining 15 days.
We are looking for nature, snorkeling, some local atmosphere, and would
prefer to stay away from the more crowded locations and AI resorts. Our
ideal place would be a kind of cottage or small hotel beach-front with
some coral reef to explore... is it still possible to find such places
and organize the trip via the Web? Since it's a honeymoon trip, it
would be better to have everything settled in booked in advance....
Until now we are considering Antigua, Anguilla or Turks & Caicos, any
other idea?
Thanks in advance for any suggestion.
Aldo