Some stuff I (mostly) haven’t seen mentioned yet:

• Check out the schedule at the New England Conservatory (necmusic.edu) - they often have student and/or faculty recitals for cheap or free. If you can find anything happening in Jordan Hall, I’d highly recommend going to it, as it’s a gorgeous, historical performing venue (google “nec jordan hall” for images); note: the NEC is a top-tier music school in the country (oldest music conservatory in the USA), so you might find that worth touring around.
• the Sanders Theater at Harvard is also a nice, “historic” theater, and has some good events; not usually cheap, though. I’m not sure which is actually smaller, but Sanders feels smaller than Jordan
• Harvard Square has been getting bought out by big money, and has lost most of its local flair. Some remaining local places: Black Sheep Bagel, Orinoco (Venezuelan food), Club Passim (live mic/folk, many events), Shays (bar/pub), Whitneys (bar/pub), Charlie’s (pub/restaurant; home of the double cheeseburger!)
• Harvard Square not strictly local but long-standing institutions: Felipe’s (mentioned in other comments), El Jefe’s, LA Burdick’s (chocolate cafe—get their dark hot chocolate! Or any, but the dark will drop you)
• Harvard Square non-food: The Coop (Harvard’s official bookstore, not fun anymore but the sell the Harvard paraphernalia all tourists want), The Harvard Bookstore (independent; not part of the coop; many events, check their calendar!)
• Harvard Square Other: explore the Smith Center and appreciate their relative new renovation with the “living plant walls” and interior outdoor atriums, explore the funky shops/eateries in the Garage (though now it’s mostly Newbury Comics on the second floor; I like the Vietnamese and Indian/Nepalese places on the first floor, though); I didn’t consider/mention spendier places like Harvest (tends to be where visiting dignitaries seem to eat)
• Harvard Square outskirts: Longfellow House and Brattle Street Mansions.. walk by old Tory money, including the house George Washington commandeered to use as his command center for the Boston Siege (operated by NPS, tours; but you can always explore the grounds including a very nice garden in the back); Mount Auburn Cemetery - first “garden” cemetery in the US; served as a model for intentionally landscaped cemeteries worldwide; really pretty grounds with a gothic-looking(?) chapel and also a tall hill in the middle with fantastic views all the way to downtown Boston. If it’s in season (April-October), you can get an even better, more 360 degree view by climbing Washington Tower on top of the hill. Longfellow House is probably a 10 minute (or less) walk from the Harvard T. Mt Auburn Cemetery is a bit further, but still walkable, less than 30 minutes (and it’s a scenic walk down Brattle/Mt Auburn). Note: if you come in mid-late April, on Sundays (and maybe Saturdays..we’ll see if they keep doing that in 2023) they close the busy, multi-lane “closest thing to a highway in Cambridge” street called Memorial Drive. Not the entirety of it, but a good chunk, with Harvard’s campus being mostly in the center of the closures. It becomes a pedestrian park.
• If you’re a gamer, see if you’ll be here during PAX East and consider picking up badges for that huge event