Quincy Market for lunch & shopping. Lots of food booths surrounded by boutiques. Touristy, but still great.
Isabella Stewart Gardner Museum was one of my favorite places, not so much for the art but because the courtyard is gorgeous and they have a delicious brunch.
Newbury and Cambridge are the hip areas with lots of college students.
Also, I don't know if you're driving, but the towns I like best in MA are Amherst, Stockbridge, and Salem. They're very New Englandy, have lots of thrift stores, and aren't as dirty as Boston. Also, there is a tiny town called Montague that has a place called the Bookmill. J Mascis and Thurston Moore sometimes hang out there and jam with the locals, according to my cousin who lives there. It's a pretty hippie, folksy little town.
If you're into Weird stuff, pick up a book called "Weird New England." I always buy the "Weird" series books for any state I visit because they have all the hidden, bizarre places to check out.