Unless you plan to stay in one town or resort for the duration of your visit, a car is necessary for exploring and getting from place to place on Long Island. If you're coming from more than 200 mi away, consider flying into one of the airports and renting a car. Although train and bus service is regular, you may have to wait around for connections, depending on your destination. Taxi and car services can get you from place to place, but this is an expensive way to go if you're using them as your only mode of transportation.
Most of Fire Island isn't accessible by car, so plan on taking a ferry. If you drive to the ferry, park your car in one of the lots by the boat terminals. Alternatively, consider taking the Long Island Rail Road to a South Shore village with ferry service (Sayville or Patchogue, for example). Shelter Island also is reached by ferry service, but you can bring your car aboard. Shelter Island ferries leave from Sag Harbor and Greenport; in Greenport, the terminal is a short walk from the LIRR station.