The island is small — you can drive around it in a couple of hours — but public transport is patchy. The cheapest option is the network of shared minibuses ("HA" plate vans) that run a circular route around the main island; they are crowded, music-blasting, and an experience in themselves, but they don't go to the southeast peninsula resorts.
Taxis are plentiful but expensive, with fixed government rates posted at the port and airport. For independent exploring, a rental car is the easiest option — but note that St. Kitts drives on the left, you'll need to buy a temporary local driving permit, and the road around the north of the island has hairpin bends and the occasional wandering goat. The Dr. Kennedy Simmonds Highway down the peninsula is one of the best-paved roads in the Caribbean.
The Scenic Railway loops the perimeter of the main island in a three-hour sightseeing trip and runs when cruise ships are in port — touristy but a relaxed way to see the coast.