Don’t bring your car over if you’re just coming for a short stay. It’s outrageously expensive and gas on the island is far from cheap.
There’s places by both ferry terminals that you can rent bicycles from as well as bus terminals that you can hop right on. You can get a day pass, 3-day pass etc. for a very reasonable rate and the buses run all day and pick up at most posts every half hour.
DON’T RENT A MOPED. They are so unsafe and with the way the roads/lack of traffic signs is – there are frequently moped accidents. Do the people who live there and the people driving a favor – just don’t rent them, lol.
Get over to Chappquiddick on bike if you can, enjoy Edgartown in the evening when they light up the streets. Don’t go to Mad Martha’s … the people who work there are generally quite unfriendly. Try Scoops instead. Catch dinner @ The Wharf, Seafood Shanty or Nancy’s in O.B. <—upstairs, on the deck with the pick up window – food is MUCH better (albeit a bit pricier) upstairs.
Jump off the Jaws Bridge along State Beach between O.B. and Edgartown – if you have kids, swim @ State Beach closer to Edgartown (by the lifeguard stands – best sand quality and water depths, IMO) – if you have older kids, teens or are with just adults – head to South Beach for some great waves and good atmosphere.