I can help ansser this one, I've been on about 25 cruises, and most recently was in the Bahamas last week on a 7 night where we stopped in Nassau.
General rule on cruises is bring half as much as you think you need and twice as much cash

The most I ever pack is one suit case, a carry on and a backpack, that is even when I go for 2 weeks - it's so easy to overpack. Best items to bring are things that do not wrinkle easily.
Sometimes if you bring strickly carry-on luggage you can get off the ship first when you return. I've done that on short cruises.
Definitely comfortable shoes to walk in at ports, also if seas get rough you'd like a flat comfy pair of dress shoes, no one is looking at your feet if the seas get rough. Don't expect it to be rough but do prepare.
On a 4 nigher its mostly going to be a party crowd, not going to see many tuxes and gowns etc. Coctail dresses, and dressy pant suits are fine, and usually only needed maybe 1 or 2 nights, the other nights should be casual wear, usually no shorts in the dining room.
You only need a small purse for on-board, to carry your cruise card and some cash if you are off to the casino etc.
A tote or backpack is nice for ports and shopping - sometimes the cruiselines provide a tote.
A highlighter pen - so you can highlight the activities on the daily schedule for which you'd like to attend.
For a tasty treat to take home check out the Tortuga Rum store, its main operations are in Grand Cayman but they have one there - my favourite is the Pineapple one

nice gift to bring home.
I've only been to Nassau, they had some nice shops not a lot for bags, Solomon's Mines was good though if you get a map or shop talk onboard maybe look that up.
The Atlantas Resort at Paradise Island is very cool, I had my favourite meal ever at a place called Bimini Road on the resort - there are also high end shops in the hotel, I just zipped by them.
If you are going to Freeport, I have no idea, haven't been there.
Here's a cut and paste from Celebrity Cruises' site, my line of choice, if you are on Carnival, Royal Carribean or another liine check out their website for a "planning for your cruise" section .. this is what Celebrity's site says:
***************
To better help you prepare for your vacation, here are some guidelines on what to pack. Keep in mind that 'relaxed elegance' are the bywords for dress onboard each of Celebrity Cruises' ships. And though your own taste, interests and destinations will determine what you pack, you'll need three types of clothing: casual shipboard attire or day wear, comfortable apparel for sightseeing and evening wear.
What to Pack
Whether on the ship or in a port of call, stylish resort wear is fine for both men and women. For warmer climates, this includes walking shorts and polo shirts, swim suits and cover-ups, lightweight slacks or skirts, and a light sweater. When visiting cooler climates, casual clothing that can be easily layered for warmth or comfort is recommended, along with a raincoat or umbrella, waterproof hat and gloves.
On all Celebrity cruises, it's wise to bring along comfortable, rubber-soled shoes, hat, sunglasses and a good sunscreen. Remember to pack any prescription medications or other personal care items you'll need.
Evening Attire
Evenings aboard Celebrity include three types of dress: formal, informal and casual.
Formal attire calls for an evening gown or cocktail dress for ladies and a tuxedo, dinner jacket or dark suit for gentlemen. Elegant, formal evenings include the Captain's Welcome Cocktail Party and Dinner, the Captain's Farewell Dinner, and a third or fourth "formal" night for those cruises sailing more than seven nights.
For your convenience, arrange for tailored, elegant formalwear to be delivered directly to your stateroom with Cruiseline Formalwear. With an elegant line of eveningwear for ladies and classic, stylish formalwear for men, you'll look as famous as you feel.
On informal nights, ladies may wear an informal dress or pants and blouse, while gentlemen may wear a jacket and tie with slacks. Casual nights call for casual, comfortable wear, so women may opt for a pantsuit or sporty outfit while gentlemen may choose a sport shirt and slacks.
The daily program, delivered to your stateroom and available at the Guest Relations Desk, will be your guide to the appropriate attire each evening. Dining in jeans, t-shirts, shorts, tank tops or swimsuits detracts from the overall ambience and is therefore not allowed in the main dining room after 6:00pm.
A casual dress code applies at all times onboard Celebrity Xpedition.
Baggage Tips
Although there is no baggage limitation aboard the ship, for your own comfort, we recommend that you limit the number of pieces you bring. Guests with air arrangements booked through Celebrity Cruises are usually permitted two pieces of checked luggage (not including oversize bags and sports equipment) and a carry-on bag. Check with your air carrier for specific restrictions.
Bring an overnight bag for your last night at sea. Staff will collect your bags the night before you leave the ship, and you will need an overnight bag to carry your night clothes and toiletries.
Be sure to identify all your bags, including carry-ons. Use your Celebrity Cruises luggage tags, which should be filled out and affixed to each piece before you leave home.
Insure all luggage and valuables. Celebrity Cruises cannot be responsible for their loss or damage. Baggage protection, as well as a variety of other coverages designed to protect you during your cruise vacation,
Check to see that your tickets, boarding card and other documents are in order, and pack them in a bag that you carry with you. Your Embarkation Card and Celebrity Seapass Account application should be completed before you arrive at the pier.