Tour art museums, perhaps on a First Thursday Art Walk in Pioneer Square, or take classes in media such as glass-blowing classes at the Seattle Glassblowing Studio or classes in various media (including glass) from Pratt Fine Arts Center.
The Kirkland Gallery Association features special artist receptions and rotating shows the second Thursday of each month from 6:00 to 9:00pm.
Arts festivals Edit
There are several arts festivals held in the Seattle area, primarily in the summer months.
The NWSource.com Summer Brewfest at St. Edward State Park in Kenmore on Father's Day weekend (in 2005, June 17-19) includes some arts and crafts booths, as well as musical and other live performances.
Fremont First Friday Art Walk is held the first Friday of every month from 6-9pm with most of the participants providing food or music or an event of some kind. While the better showings tend to be at the galleries, some of the other places can pull in very talented artists. Lots of diversity (some abstract, some realistic, some paintings, some sculpture), lots of interesting art, many quality shows, maps available at most places, most places put out a sign so they're easy to find. The Edge of Glass gallery does sculpting demonstrations and you can sign up to receive one for free, the only catch is you have to stay and watch the artist (James Curtis) make it for you. More info at 
Edmonds Third Thursday art walk is also 6-9pm. Many of the galleries provide food or music. Very pedestrian friendly neighborhood, flowers everywhere, galleries close together and well marked, maps available most places. Semantics, the Francis Anderson Center in particular worthwhile stops. More info at 
Ballard Second Saturday is worthwhile but has a more commercial feel and most places don't have maps. Some shops stay open later, but it is officially 6-9. Still definitely worth going, fun neighborhood. More info at 
Pioneer Square has First Thursday. More info at www.pioneersquare.org/first_thursday
Issaquah has a quarterly art walk. Pretty neighborhood, more of a craft focus to the art. Make sure you have the current schedule before making a trip.