Martha's Vineyard is a magical Island broken up into several small towns, each with its own separate feel and its own separate attractions and things that make it special. The whole island itself is kind of small, and has a real community feeling. The people are friendly and willing to give you lots of advice on good places to go to eat and to stay. I'd recommend you stay in Aquinnah because it has the most beautiful views of any part of the Island- the red bluffs are magnificent at sunrise. Visit the other towns though- Edgartown has amazing shopping and is the best place to get a great (albeit fancy and somewhat expensive meal).
Martha's Vineyard is, of course, an Island so you can get your fill of water activities—jetskiing, sailing, or you can go clamming (you might not find a whole lot of clams but it is still fun). The waves are not typically very high so it's a good place to swim or sit on the beach and have a leisurely afternoon.
Of course, make sure you go in the summer—this is Massachusetts so it gets very cold in the winter and most of the tourist stuff is closed. The busy season is memorial day through September, so this is the time to go (although earlier or later in the summer might be less crowded for you if you want a quieter time).
Quick Tips:
Everyone is very friendly and nice to the tourists (tourism is one of the main sources of income on the island. The best way to figure out what to do and what to see is to ask the locals. Also, stay in a bed and breakfast if you are only going for a brief time. It's much quainter and nicer than a hotel and you'll get a much better feel of the hominess of the island. You can also rent houses by the week or for the summer if you want to—this can be great because people will want to visit you there if you rent a house, since Martha's Vineyard is such a well known and wonderful place to visit.
The island is, in general, a casual place although if you go into Edgartown be prepared to dress up a little more and spend a little more money.
Best Way To Get Around:
You cannot bring your car unless you bring it on the fairy, which is kind of expensive—so if you plan to do this make sure you plan ahead and make arrangements and find out the cost. I'd recommend not doing this though—you will spend a lot of time sitting in traffic.
There's pretty good public transportation—you can buy a bus pass for like and take the bus all day.
The best way to get around by far, though is biking—it's a small enough island that you can get almost anywhere by bike and its a really great way to experience the island and enjoy the outdoors. You can rent bikes there, or bring your own if you want to (although if you fly in, might be hard to bring your bike on the airplane).