The most accessible beach on the island is Owen Park Beach, off Main Street in Vineyard Haven [1]. The strip of sand at the bottom of a grassy hill affords a view of the sailboats in the harbor.
Despite the two miles of pristine sand and gentle surf, you may feel nervous getting in the water at Joseph Sylvia State Beach along Beach Road between Oak Bluffs [2] and Edgartown [1]. After all, it was along this stretch that much of the movie Jaws was filmed.
A more dramatic landscape is Katama Beach, a narrow barrier beach near Edgartown, with surf crashing on one side and a warm salt-water lagoon on the other.
Lobsterville Beach, near Aquinnah, is a favorite for beachcombing, with plenty of shells and sea glass that wash up on the shore. Bring your bike, however, as there is no parking along the road.
Links:
[1] http://www.moon.com/destinations/new-england/massachusetts/cape-cod/martha-s-vineyard/down-island-sights/edgartown-and-vineyard-haven
[2] http://www.moon.com/destinations/new-england/massachusetts/cape-cod/martha-s-vineyard/down-island-sights/oak-bluffs