T.F. Green Airport (PVD) in Providence / Warwick and many other regional airports serve air travelers from all parts of Rhode Island. Westerly State Airport (WST) and Block Island State Airport (BID) off the coast also have airports serviced by regional airlines.
T.F. Green is a small airport, but it serves most of the major airlines and is conveniently located in Warwick, just 15 minutes from Providence and 35 minutes from Newport.
Depending on where in Rhode Island you plan to visit or are departing from, it may benefit you to fly into or out of Logan (BOS) in Boston, MA in order to fly direct and save yourself a bit of money or a transfer.