June is a great time to go biking along the Charles River. Another good bike ride is down the Southwest Corridor from Back Bay to Forest Hills. Before reaching Forest Hills, you can detour down Williams Street to Franklin Park and ride around there. Forest Hills Cemetery is also nice for a ride. It's a Victorian cemetery with tons of sculptures and a pond.
Definitely check out restaurants in the South End and Jamaica Plain. Faneuil Hall is full of tourists.
In terms of sites, there's plenty to enjoy if you like art, history, and architecture. Boston has everything from the colonial era to the present day and there's a lot to see in the neighborhoods as well as downtown.