On Nassau, you can get a cab over to Paradise Island which has nice beaches. Your costs will be the cab ride (which may be negotiable), and perhaps a fee at the beach if you want to rent a beach chair or get something to drink.
On St. Thomas, ditto. A cab will get you to Megans Bay or Sapphire Beach. If you want an excursion here and like snorkeling, consider the trip to St. John. The "excursion" is basically an open air bus ride, but the beach (a national park) is very pretty, and--at least on early excursions--not very crowded. Our favorite excursion in St. Thomas was sailing to Buck Island for snorkeling. The boat held only six guests, there was no tour talk, and the captain let interested passengers take the wheel and get the feel of sailing the boat. Pricey, but well worth it.
On St. Maarten. Once again, great beaches are a cab ride away. Orient Beach is a "clothing optional" beach so be aware of that if nudity is a concern to you. The beach is kind of wild (in that the waves are rougher and the beach is dotted with grasses and small dunes) but very nice. There are also beaches on the other side of the island., which are flatter with calmer waters. These have more water sports available--windsurfing, parasailing, and the like. All are attached to hotels, but there should be no charge unless you want to rent a chair and/or umbrella.
Tips for taxis. Arranage with the driver to pick you up in enough time to get back to the ship or into town for shopping if that's what you want to do and promise the tip for the return ride. We have never had a problem with taxi drivers coming back (which I believe they would do even if you tip them both ways). In any case, there are alwasy plenty of taxi drivers around.
The ship will provide you with basic outline maps showing the main points of interest and which may also include info about taxi fares and whether negotiating is expected or accepted. Many ports have taken to posting taxi fares on a board near the ship and have staff to round folks up who are all going to the same place. For the beach: Take towels, sunscreen, and water with you. Leave expensive cameras, jewelry, and anything you would really mind losing on the ship. Invest in a waterproof case you can wear securely around your neck or clipped to your suit to hold cash, a photocopy of your passport, your ship security key card, and a couple of traveler's checks or one credit card if you feel you'd need one.
..Joanne