The Amtrak Downeaster goes to Portland, Maine though a roundtrip is a long day. There is a bus from Boston airport/train station into New Hampshire. For example, it is 1.5 hours on the bus to Concord, NH (cute town and state capital) and has stops iin-between that might possibly be of interest.

I do like Providence. There is a nice downtown area to walk around, and it is on the water. The Rhode Island School of Design is also there (RISD), which as an art/design museum. The Providence Public Library was renovated not long ago and might be worth a walk through. Federal hill neighborhood has various restaurants.

The Boston/Cambridge area has a lot of stuff and so you could just stay put. If you're into history, art, politics, neighborhoods, you'll have plenty to do and no need to venture out.