There are a lot of well maintained bike paths. They sort of go around the island. And through the woods. In the nice weather the bike paths and the roads are packed. Be patient on the bike paths.
Google " Martha's Vineyard bike map". Then just google or use links to learn about anything.
Last time I was there they were giving them out at tourist/bathroom building at the far side of the parking lot when you get off the boat. There are lots of things like that. The road on Chappaquidick turns to soft sand,after a while, almost not passable on a road bike. Plenty of sliding around on an MTB. At the end of the road there is the bridge Kennedy drove off of. There is a big beach with 4wd truck tours. You can walk your bike on the sandy areas of that road.
Ask about the "Campground". Lots of gingerbread small cottages. Not a campground any more. Plenty of info on the island. You can ride to everywhere from Edgartown.