Pro Mountain Sports in the University District (University Way and about 60th), Feathered Friends across from REI in downtown Seattle, Marmot Mountain Works in Bellevue, across the street from Bellevue Square Mall (NE 8th and Bellevue Way, down stairs) are the best stores for technical gear, soft goods, and clothing. If you go to Feathered Friends you may as well stop in at the REI flagship store across the street, if for no other reason than to admire the architectural use of OSB panels and other inexpesive building materials in unique ways and a whole lot of women's clothing. Also, the mural on the Feathered Friends building was done by former Seattle local Andreas Schmidt is really cool, who is rumored to have climbed more routes in Selected Climbs in the Cascades than any other climber.
Don't forget Outdoor Research down on 1st ave south. Not climbing gear per se, though they do have that as well, but they will generally have your gloves, hats, when the other stores do not as they only carry them in the winter time.
Since not everyone has $$$ to simply buy the lightest piece of gear that dies after 3 uses, here are a couple options one might not generally think of in the Seattle Area:
If you want bomber work jacket/pants or even for just camping, go by Filson on 4th avenue south. Also on 1st avenue south are a couple contractor supply places that carry work clothes. IE pants made from Cordura nylon which will take Serious abuse, but are heavy.