Harvard Museum of Natural History is awesome!!! It's free to Massachusetts residents every Sunday morning (year-round) from 9:00 am to 12:00 pm and on Wednesdays from 3:00 pm to 5:00 pm (September through May). Proof of residency required. RISD museum in Providence (which is also awesome) is free on Sundays all day and Thurs / Fri after five - just go to their website and reserve a ticket before you go.

Also, if you have a ConnectorCare card or an EBT card, you can get in to many museums for a steep discount or for free.