Edgartown Martha's Vineyard...
A few more... all in Edgartown: For pub food and atmosphere- Newes From America in the Kelly House and David Ryan's (downstairs). For breakfast- Among the Flowers Cafe or get a coffee and cruller from Humphries and walk to the town dock to watch the boats come and go. More upscale but still casual? Lattanzi's Pizza (they have a lot more than pizza). splurge? The Grill on Main (best value) or Detente. All of these are fairly small and serene- perfect if you are alone.
I live here year round, and have seen lots of places come & go. The Saturday farmers market in West Tisbury is great, can be a little expensive. For breakfast or lunch, in Vineyard Haven try the Artcliff Diner, in Oak Bluffs Slice of Life and in Edgartown the Right Fork Diner. BEST fried anything is the Menemsha Bite. Dining alone anywhere should not be a problem, I usually bring a book or just people watch. Have fun!!