If you have never been to any island in the Caribbean, I would suggest starting with one that is fairly easy to get to and allows you to do a bit of island hopping. Also, look for condo or villa rentals so you will have a kitchen to keep the food expense down.
One suggestion is St Thomas (cheaper accommodations than St John) but with the ability to visit several other islands. I happen to think the easiest snorkeling and diving are available in the Virgins.
My first Caribbean experience was St Maarten-Saba-St Barts in 12 days. French, Dutch, mountains, flats. You could also put Anguilla in that mix. It has been many years since I was in Cozumel and Cancun, but they had great beaches and snorkeling at one time.
These three areas are kind of entry destinations. You can get a taste and see if you like it.