Go sightseeing and check out historic sites. Go to quincy market, visit Boston Commons (public park). Museum of Science or Fine Arts are both very popular destinations for tourist.
South End used to be the gay neighbourhood of Boston and still is in many ways. So walk around there, check out the businesses, art gallery stores. You can also visit Back Bay's Newbury Street which houses many restraurants, bars, and high end retail stores.
I rarely go exploring downtown Boston, so I hope this info can help.