The Martha's Vineyard Transit Authority makes the island very accessible for tourists.  Although I'm sure locals take the buses, too, the VTA, as it's known, makes it very easy to get around the island cheaply and conveniently.

Routes

They offer both transportation all around the island as well as shuttle service into and out of some towns where parking can be scarce (e.g., Edgartown service from near the triangle shopping district into the central historic district where parking is virtually non-existent).  They have routes that make it easy to visit one town from another or to visit further-out beaches (such as South Beach outside Edgartown) or even lighthouses and landmarks (like getting to Gay Head/ Aquinnah to see the cliffs).

Fares

Fares are $1 per town each way (including the town of origin) so this should net out to $2-5 for just about any route on the island.  Exact change is required, although passes are cost-efficient and alleviate the "exact change" issue if you're on and off a few times in a day or a week.  Passes can be purchased on the bus, at the Steamship Authority or at select ticket sellers named on the VTA site.  Discounts are also available for seniors, those with disabilities.  Children under the age of 6 travel for free with a fare-paying adult. 

TIP:  It's worth noting that the VTA bus stops in some towns may have a shop for traveler necessities (sunscreen, poncho, postcards, cold drinks) and may be co-located with the only public restrooms in a town so even if you don't find yourself riding, you may find it valuable to find them on the map.  In all cases, your bus driver can direct you to these services and -- frankly, public restrooms are typically scarce -- so it's worth asking when you're dropped off in any town you visit.

Notes

The buses themselves are large, white, modern buses with big purple VTA symbols.  Route maps and more information are available at their website.  Bikes can be transported as well, although there may be certain routes or certain buses; plus there may be an additional fee.

Links to off-site resources

Martha's Vineyard Transit Authority (VTA)  www.vineyardtransit.com