I suppose we could answer most questions by recommending a search engine, but it wouldn't make much of a forum, would it, so I'll try and be a little more helpful with information about the islands I have visited.
Generally, most islands will have taxis on the quayside and there is often a despatcher who will make up parties so the cost is shared. Agree the fare before travelling. So it is possible and simple to take your own trips, although if you want somewhat specialised stuff like snorkelling, it is preferable to take the ship organised trips.
Tortola - Ship docks close to the town centre and "capital" of the island, Road Town, which is a small town with a few shops and very friendly locals - a pleasant place for a stroll. For a beach trip, try Cane Garden Bay on the north of the island over a short but mountainous road with great views.
St Maarten - ship docks near to the "capital" of the Dutch side of the island, Philipsburg. There is a local ferry boat to the town centre for a few dollars. Town is pleasant and safe with a good beach. But I can recommend a taxi tour of the island. This will enable you to see the French side of the island and its main town of Marigot. The highlight of our trip was a visit to a nudist beach, which seemed to be for the benefit of our driver rather than his passengers!
St Lucia - again within walking distance to a pleasant town, Castries. The Pitons can be visited by taxi.
Martinique - dock close to the town centre of Fort de France, a very large town by Caribbean standards and very French! Plenty to explore in the town, but you can do your own thing by taking a public ferry to Pointe de Boute where there are good beaches.
Barbados - the dock is a short way out of the main town, Bridgetown, but you can walk into town where there is a beach.
I am not sure if that is meant to be a snide comment or not - but the site I directed Mr Lloyd to has literally thousands of contributions by fellow cruisers, that will give him far far more information and insight into what to do on each island, and which private tours to take etc, than the information that is given here by the 4 or 5 regular contributors. Try it and see .....
We did a similar cruise earlier this year. One of the best value excursions we have ever undertaken was in St. Lucia with cosol tours (google it). It was full day, with breakfast, lunch, and never ending supply of beer and soft drinks. It was 50 dollars each...and even with the pound devaluing will still represent terrific value. What he says you get..on his web site...is what you get and more. Cannot speak to highly of him. Look him up on cruise critic...he gets fabulous reviews. If you are going to St. Maarten and like aircraft, think of going to Mayo beach for a very unusual experience. The beach and bar are at next to where the planes land....they come in six foot above your head!!!! You can see this on youtube. Any more info. please let me know.
Have just done the same cruise basically.
St Lucia - the helicopter ride over the pitons and rainforest Tenders in the main town centre
Barbados- the sunset cruise . 15mins from main bridgetown
Tortola - go to sugar cane bay beach $10 by taxi (absolutley stunning) 10mins from beach
Catalina island you will be tendered to the beach for a barbecue
Bonaire - 5 min walk in to centre
Martinique - 15mins walk to capital
St maarten - 10min by taxi to centre
Aruba - $10 taxi ride to eagle beach as you will bert in the main port which is just a industrial estate
St Vincent- berthed in centre as only very small.