Here's a starter overview of art museums and galleries worth visiting:
The Frye Art Museum
, on Capitol Hill, is my favorite museum in the city (the building itself is beautiful), always something intriguing and often it stretches me far. The Frye is always free, by its charter.
The Henry Art Museum
on the University of Washington campus in the U District is the best venue for modern art. They too always have provocative shows. The Henry is free on Thursdays.
The Seattle Art Museum
has three venues. Well... actually right now it only has one, as the main, downtown museum is being remodeled (thank god), and the new sculpture garden is not yet open. None-the-less, the Asian Art Museum branch is open and is a fabulous building in Volunteer Park on Capitol Hill (worth a visit just for the view). Admission to the Seattle Asian Art Museum is free on the first Thursday and first Saturday of each month. The Sculpture Park branch will open in late January and will always be free. The main museum will reopen in May.
There is a slew of great art galleries in Seattle. Although they have traditionally centered in Pioneer Square (and there is still a density of galleries there), other neighborhoods are growing their own “gallery alleys”, including Belltown, Ballard, and Georgetown. Here are links to some of my favorites:
The Davidson Galleries
The Seders Gallery.com/
There is an art walk the first Thursday of every month in the Pioneer Square neighborhood from 6—8pm. Galleries open new shows that evening, often with snacks and wine. Occidental Plaza is always filled that evening with new/young/unrepresented artists selling their wares.