See the information on the Long Island Railroad at mta.info
Montauk is about 3 hours. Montauk is all the way at the end of the island. Other places like East Hampton are a little closer on the same line. The fare is $25 one-way peak, $18.25 off peak. (You might be able to bring that down slightly by taking LIRR part of the way and then switching to local buses but the extra time involved will almost certainly make that not worthwhile.)
Are you looking for anything in particular - beaches, rich people's houses, parks, downtowns?