The Space Needle and Pike Place Market are usually at the top of every tourist's 'must-see' list.
The Space Needle is located in the middle of Seattle Science Center, with a number of different pavilions dedicated to bringing scientific concepts and applications to a broad, diverse audience. Be sure to add the science center to a trip to the Needle.
Gasworks Park on Lake Union.
Visit Fremont, a lively, very liberal neighborhood north of downtown. The Fremont Troll, located under a bridge, naturally, is an excellent example of urban art. Fremont also boasts a statue of Lenin, a not-very-subtle poke at Fremont's left-leaning politics. Someone painted the statue's hands blood-red to remind visitors that Lenin was no squishy, kum-bah-yah liberal. There is also 'Waiting For The Interurban', a cluster of statues representing light-rail commuters at one of Fremont's streetcar stops. Half a block away is a statue memorializing J.P. Patches, a beloved children's television clown and variety show host. It depicts him and his sidekick running to catch the aforementioned interurban.
In a city famous for the arts, there are so many art museums, professional and community theaters, and art festivals, you'll never get a chance to visit them all. Read 'The Stranger', Seattle's progressive underground newspaper, to help you plan art outings.
Seattle is the birthplace of Starbuck's, but since a Starbuck's in Seattle is little different than a Starbuck's in, say, Budapest, stick with the smaller, local, family, or cooperatively-owned coffee shops.
The famed 'Seattle Underground Tour' is a bit of a tourist trap, but it's fun, and an interesting way to learn about Seattle's early history. It originates in a bar near Pioneer Square, (the 'Square', by the way, is triangular...)
Woodland Park Zoo, erstwhile home of the world-famous Bobo the gorilla.
Head across I-90 (over the world's largest floating bridge) to Bellevue for excellent high-end dining and shopping. Bellevue has been named the Most Walkable City In America, and so naturally is very pedestrian-friendly.
If small clubs for listening to up-and-coming bands is your thing, I recommend the Nectar Lounge, The Tractor, The Crocodile in Belltown, and the Triple Door.
Travel up to Kerry Park on lower Queen Anne Hill, and face south for the 'Frasier View', a close approximation of the view from Frasier's condo on the self-titled sitcom. (The view shown on the 'Frasier' set is actually a composite photograph.)
Broadway on Capitol Hill is the epicenter of Seattle's thriving gay scene. Lots of great restaurants, shops, and clubs.
That would make for a good start... 
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