Many hotels and resorts offer tennis courts. There is good hiking in the islands - explore the active volcanoes in St. Lucia or Dominica. Try some of the nature parks. Rich in history, discover the Caribbean and learn about Caribs, Columbus, the pirates and buccaneers and the slave trade all of which have shaped the people and culture of the islands. Wonderful historical sites abound.
Music is at the core of the Caribbean culture.
From salsa in Puerto Rico and Cuba, merengue in Dominican Republic, zouk in Martinique and Guadeloupe, calypso and soca in Trinidad to reggae immortalized by Bob Marley in Jamaica and heard on most of the islands. Any of the Caribbean Carnivals are worth visiting. The famous Trinidad carnival in February each year attracts visitors from around the world. Eating out in the Caribbean is to be recommended. Don't forget to check out the local rums and beers.