Needless to say there are a number of possibilities for a Day Trip to a LI Beach via the LIRR.
A few more to consider, take the morning train to Greenport, then the quick ferry ride to Shelter Island. A short cab ride will bring you to Cresent Beach and what has become "the scene beach of the East End", the section of Cresent Beach in front of the South Beach Hotel.
For a more serene beach take a cab ride to Wades Beach on Shelter Island, and it has facilities.
Montauk is as mentioned, unique and well worth the trip. Take the morning train out and then have an early lunch at Duryea's Lobster Pond, a short walk from the station, and then cab it to Ditch Plains Beach.
For a true taste of the Hamptons, take the train to East Hampton, and then cab it to Main Beach (it is too far to walk). This beach has full facilities, and it is ranked as one of the top 10 beaches in the USA year after year. Yes it is the Hamptons, at it most "Hamptons", but is beautiful.
One word of caution, if you plan a day trip on a weekday to The East End,
LIRR service remains limited. Sometimes the return trip (on a weekday) is best done by the far more frequent Hampton Jitney.
www.hamptonjitney.com
Ken