Accessed by frequent ferry (the 'Chappy Ferry') from Edgartown harbor, Chappaquiddick is a small island with a big history, thanks to a fatal accident in 1969 that involved Senator Ted Kennedy. The island's population is around 180; it's a popular spot with visitors for its beaches, cycling, hiking, fishing and birding. Bring a bike and a picnic over on the ferry and go exploring (there is one shop and no restaurants). Mytoi, a beautiful Japanese-style garden, is free and worth a visit if you're in the area.
It's worth looking into the summer tours of Cape Poge Wildlife Reserve, offered by a nonprofit group called the Trustees. You must book ahead for tours that visit the remote lighthouse or enjoy naturalist-guided kayaking on Cape Poge Bay.