Are you interested in the Fine Arts? The Boston Museum of Fine Arts is a truly world class gallery. Also the Isabell Stewart Gardiner museum although excentric is certainly interesting. Then you have the Fogg, Sackler, Busch-Reisinger
Museums at Harvard, in Salem (a short train ride from North Station) is the Peabody Essex Museum. Also lots of public art (for example the Robert Gould Shaw Memorial is across from the State House on Beacon Street near Park).
Also for Science the Harvard Museum of Natural History and the The Harvard Museum of Comparative Zoology (The Agassiz Museum). The Boston Museum of Science is supposedly quite good (but it isn't my cup of tea).
Lots and lots of History, from the three Burial Grounds (Copp's Hill, King's Chapel and the Granary), to Christ Church Boston (Old North Church) King's Chapel, the site of the "Boston Masacre", Paul Revere's House, the U.S.S.
Constitution the second oldest warship in Commission (after H.M.S. Victory) and the oldest afloat - something happened or someone lived at practically every third building.
When you are at Harvard the oldest College (as in University north of Mexico) Massachusetts Hall is the oldest extant college building in North American; and down the street on Cambridge Common, George Washington took command of the Continential Army.
