Boston and Cambridge are amazing cities with a lot to do, and if the weather is nice you can have a great time just walking around downtown Boston and enjoying America's Walking City (but not according to those fakers at the Brookings Institute).
Some sites we love within walking distance of Copley Square:
Newbury Street (fancy and budget shopping, cafes, excellent people watching)
Boston Public Garden (where the famous Swan Boats are)
Faneuil Hall (tourist trap, but nice to walk around and get a view of City Hall)
Other sites further afield in Boston:
Isabella Stewart Gardner Museum
Or...you can hop over the bridge to Cambridge (Our Fair City) and see some great stuff. Sure, there's Harvard Square, but why not try something new?:
Harvard and MIT campuses (well, MIT campus is maybe not so great, but it does have a new and controversial Frank Gehry building...)
Harvard Museum of Natural History (the glass flowers are unique and incredible)
Central Square (the one and only)
Charles River Esplanade (a nice walk from Boston to Cambridge and back)