Having worked on ships for 12 years--3 years just here in the Caribbean, I got a few recommendations. I've been to all the islands you listed...many times. Too many for a few of them. I will list some suggestions, but in no particular order of importance:
1. Stay away from gambling in the Casino.
2. Be prepared to tip bartenders, service/chamber maids, waiters, cocktail waitresses, Spa/masseuse workers, etc.
3. Be prepared to eat, sleep, eat again, drink, then eat again. Wake up, and repeat.
4. Excursions/Tours are very expensive. Some suck big time, some are very interesting. Depends on the island and the tour company, etc. Be prepared to read in-between the lines of which tours are worth it, and which ones are a waste of time.
5. Bring condoms. The waiters, bartenders, officers, shop attendants, Casino dealers, cooks, security officers, deck crew, and everyone else hidden deep below the ship are running around horny and are willing to have sex with anything that moves or bends over. Everyone is locked up and humping anything stationary.
6. Bring some formal clothes for formal nights. Typically, for a one week cruise, there will be two formal nights where it might behoove you to dress up, or look like a redneck. However, here in the Caribbean, these nights are not so important...it depends on what cruise ship/cruise line you are on. Some cruise lines are very high end/extremely expensive and therefore attract a wealthier crowd...they tend to take formal nights very, very serious. The lower end cruise lines--Carnival, etc., not so much.
7. If you're married/faithful type, bring a good vibrator and lots of batteries. Batteries are very, very expensive in the gift shop on board the ship.
I'll think of more suggestion later.
Frank