My college roomie and are headed to Seattle in late March (she has a conference!) On my list are 2 non negotiables; The Space Needle and the Pike Place Market.
What else is a must do, we have Fri and Sat with an eaarly flight on Sun.
Chihuly glass art museum is a must see too.
Edit: It is right next to the Space Needle so not too hard to fit in.
The Experience Museum of Pop Culture is at the Seattle Center as well. While you are at Pike Place Market and the weather is decent, you may want to go down the stairs to the waterfront and walk around. The Aquarium is cool and Elliot's Oyster House or has great seafood.
I was there last week. We only went to Pikes Market - it was pouring rain. The ferry ride was great. My kids moved up there a few months ago so I'll be glad to find out some news things to do.
I really want to hike around Seattle when the weather is better. Lovely old buildings.