The Chazen is excellent museum right in the middle of the UW campus. The brand new museum is bright and airy with a wide range of mostly modern art from North America and Europe. Some of the artists are relatively obscure, but don't miss the painting by Warhol in the new building or the Vasari in the old building.
Admission is free and they have a relatively strict bag policy, but they also have free lockers to store backpacks and large bags. Great place to spend an afternoon or find a quiet place to study.