I second the RISD museum - it's a world-class museum. That whole part of Providence where the Brown and RISD campuses are is very artsy and cool.

Definitely see some of the beaches and do some coastal hiking. Either on the East Bay Bike path or at one of the wildlife preserves - Ninigret or Point Judith - there are many.

Newport is amazing. The Breakers is probably the most spectacular, but they're all impressive. Newport itself has a lot to do - dining, window shopping, etc.

Rhode Island also has some pretty, New England-y spots like Wickford and Westerly. They can be fun to walk around and have lunch.