There ain't a lot in Little Compton (a zoning thing). But I second the Boat House and Evelyn's, both in Tiverton.
Provender for a nice lunch or take out sandwiches (at Tiverton Four Corners)
Gray's Ice Cream (across the street)
Milk + Honey cheese shop for some take home cheese (also Four Corners)
Nonni's - italian - Tiverton
Westport is close to LC and Fall River is a short drive. Good Portuguese in Fall River (try Caravela's or Estoril) and New Bedford.
The Liberal Club in Fall River is a no-nonsense family style place that serves mounds of fresh seafood for super cheap. Be warned: no credit cards and they have a very quirky antiquated system of getting your drinks vs food from (and paying for) via diff waitresses.
Persimmon in Bristol gets rave reviews but I've not been. Im sure others here will suggest it.
1577 Pleasant St, Fall River, MA 02723