General Ideas
Places to go
Tour a rather unusual national park, right here in Massachusetts: the Deer Island sewage treatment plant. The other Boston Harbor Islands are great, too.
Atlas Obscura picks for the weird and wonderful in Cambridge and Boston.
If you like to bike, check out the Mass Central Rail Trail and the Minuteman Bikeway. (These run in part along old railroad lines, so are nice and flat and protected from autos in many places; they tend to be crowded with families and pets on the weekends, though.)
Did you know that there are several beaches reachable by public transit?
Harvard
In Harvard Square
Harvard Square Business Association calendar (events are not necessarily free)
@ Harvard
Harvard Gazette calendar (events are not necessarily free)
Mostly free things to do
Books and more
Anyone living in Massachusetts (including students) can get a Boston Public Library card or a Cambridge Public Library card (or both). Then get the Libby app to have access to these libraries and many others. In fact, probably the public library from your hometown is also a member of Libby, so check that out, too. Free access to so many books, magazines, audiobooks, reading guides and more!