Seattle Center and all of it's attactions (Science Center, Experience Music Project).
The waterfront, ferry rides to the penninsula, Discovery Park and old Fort Lawton (the highest population of in city bald eagles nest out there).
Pike Place Public Market and the Sanitary Market across the Street(Staurday is a bad day to go though, jam packed like sardines in a can).
Pioneer Square (though don't go down there after dark, it's an entirely different place then).
Underground tour in Pioneer Square.
Washington Mutual Tower if it's springtime, to see Seattle's most famous Peregrine Falcon pair raise their chicks (on camera, monitors in the lobby).
Take a waterfront dinner cruise (kinda pricey, but well worth it) or just a waterfront and lake tour on one of the Spirit boats (Spirit of Seattle, Spirit of Puget Sound etc).
Lots of other things to do as not, not everything is in the city of course, but there really is a lot to do there.
And if you're into nightlife and clubbing, there are quite a few places in Belltown (Downtown, though closer to Seattle Center).