There’s the corny tourist stuff that’s interesting nonetheless.
The Freedom Trail. It’s a path marked on the sidewalk that goes past about a dozen or so of the main historical landmarks, including the Bunker Hill Monument, the USS Constitution, Old North Church, Paul Revere House, Haymarket, and ending in the Boston Public Gardens near the State House. Covers a few miles and can take a couple of hours to a whole weekend, depending on how much time you spend at each place.
Harvard Square. School’s out, but it can still be an interesting place to see. Lots of shops, no real bargains. The Coop in the middle of the Square has all the Harvard souvenirs you could want (although J. August, about a block down Mass Ave, has better stuff, in my opinion). Lots of museums all around the university, including the Fogg Art Museum and the Botanical museum’s glass flower exhibit (ask for the Peabody Museum. It’s the anthropology museum, but they’re both in the same building).
The Museum of Fine Arts is always good.
Copley Plaza is good for upscale shopping, as well as some good restaurants nearby.
Newbury Street is a great place to shop and wander, with shops ranging from second-hand CDs and tattoo parlors to expensive art galleries, plus lots of unique niche shops (The Gargoyle Shop, for example). Newbury, Copley and the Museum of Fine Arts are all within a reasonable walking distance of each other.
Chinatown wasn’t the safest place when I lived in the area, but there are lots of great place to eat there. It may be better now, but my recommendation (which may be ten years out of date) is to stick to lunch time.
Try some clam chowder. Even the fast-foody places in Faneuil Hall / Quincy Market have some delicious chowder.