Biking on Martha's vineyard is great. The whole family left the car on the mainland and ventured all over MV a couple years ago (it's such a bikeable place, I spent a bit too much of my vacation puzzling and fuming about why people thought it necessary to haul all of their modern conveniences there in their enormous SUVs). I calmed down once I got on some of the shady bike paths away from traffic.
If you just want some short rides, the bike paths offer lots of pretty off-road (but nicely paved) adventures. If you want a bit more mileage, Gay Head or Menemsha are worth it. From West Tisbury (where there's a great general store for mid-ride snacks and drinks), I suggest Middle Road, as it's less traveled. I would guess Edgartown to Gay Head is about 45 miles round trip, and Menemsha about 6 or 7 miles fewer. The Bite clam shack in Menemsha is a tasty destination. We came back by way of South Road, which has more traffic -- people pulling boat trailers -- and I wouldn't recommend it.
Even with the crowds, you'll find some great peaceful rides without much trouble, and in my experience the bike paths away from Edgartown and Oak Bluffs (i.e., the western part of the island) are not only not packed, they're little traveled.