Depending on the length of your stay, you should definitely consider leaving your car on the mainland.  There are plenty of taxis to get you from the Oak Bluffs or Vineyard Haven ferry terminals to your house or hotel in Edgartown.  During peak summer season the streets and parking in downtown Edgartown are quite crowded and the downtown area is best seen on foot.

Having left the car on the mainland, those staying in Edgartown but wishing to venture further around the island have plenty of options for renting bicycles.  Most of the main roads out of town are bordered by bike paths. 

Moped rentals are also fairly popular with tourists but largely disliked by local drivers since mopeds share the road (not the bikepath) and the typical moped renter does not have a lot of experience riding one.

The Vineyard Transit Authority (VTA) offers bus service that connects Edgartown with the other island towns.  VTA busses are nicely equipped with bike racks.