Space Needle - the most expensive elevator ride in America. You can get a comparably good view for free from Bhy Kracke Park (pronounced "By Crackie") atop Queen Anne Hill.
Monorail - Not as bad a deal as the Needle. If you need to get between Downtown and Seattle Center, it's perfectly good transportation and kind of cool, but it doesn't go anywhere else. The 1962 Alweg monorail probably won't be there much longer, because it's being torn down to build a more extensive one.
Pike Place Market - An enormous working public market and popular tourist draw. Much good food to be had. The selection of fresh flowers and vegetables is excellent. And yes, they really do throw the fish around.
Museum of Flight, near Boeing Field. The name tells you right away whether you personally will find this interesting.
Troll under Aurora Bridge (Aurora Bridge is Hwy 99 as it crosses Lake Union just north of downtown, and it's legally the George Washington Memorial Bridge, but no one uses that name.) The troll is under the north end of the bridge, in the Fremont neighborhood.
Broadway on Capitol Hill, and the Pike/Pine corridor leading up from downtown to the Hill. Hipster central. In good weather, this is the best people-watching in the West. Sit down at a sidewalk cafe and watch the scenesters.
The Hiram M. Chittenden Locks' (a.k.a. Ballard Locks): boats, a salmon ladder, and beautiful gardens, just west of the built-up center of the Ballard neighborhood.
Little Saigon - centered at 12th and Jackson.
Try riding "The Tapeworm" in Philip Arnold Park in Renton, southeast of Seattle. Other trails are in this park, as well.
Take a ferry to Bremerton] and back. Almost 2 hours on the water, in a place as scenic as the Aegean Sea, walk-on passengers a little under $6 round trip.
Seattle is the home of Starbucks and of SBC (now owned by Starbucks) and Tully's, but you can do a lot better for both coffee and atmosphere. There are over a hundred good locally owned coffeehouses, which contribute greatly to making Seattle what it is. Some of the more notable are:
Zeitgeist at Second Avenue Extension and Jackson Street in Pioneer Square (+ three other locations). Elegant and arty.
Zoka, in what is variously known as the Meridian District or Tangletown, between Wallingford and Green Lake. Studenty place, great desserts, and they roast their own coffee.
Victrola, 411 15th Avenue East on Capitol Hill. Neighborhood place in a hip neighborhood.
Coffee Messiah near the corner of Olive Way and Denny on Capitol Hill, ten minutes walk (uphill) from Downtown. There is nothing like it. Religious kitsch, live entertainment most evenings, and a crowd that somehow manages to be both hip and welcoming.
B&O Espresso, 204 Belmont Ave. E. (Capitol Hill), 206-322-5028, Great desserts.
Ivar's Acres of Clams Seafood served indoors and out at a scenic downtown waterfront location popular with tourists -- eat with the seagulls! Inexpensive.
Ivar's Salmon House, north of Lake Union, a good waterfront bar and an impresive neo-longhouse interior.
Ray's Boathouse and Ray's Cafe, on Shilshole Bay west of Ballard; great views and, in the downstairs Boathouse restaurant some of the best seafood cooking in the city, priced accordingly. Upstairs, the Cafe is more casual, the food is good but not comparable to downstairs, and you can keep it to $20 a person.
Upmarket Asian fusion food at Wild Ginger (just north of the Symphony Hall) and Monsoon (obscurely located on 19th E, on the far side of Capitol Hill from downtown). Both noisy, both great.
Seattle is not known for Mexican food, but Agua Verde, on the water, just south and west of the University of Washington is a standout, attractive but informal, with creative, contemporary Mexican cooking, including a lot of great vegetarian and seafood options.
Chinese seafood restaurants are Seattle institution popular with locals, many with "live tanks". Not particularly elegant, but the food is great (if a bit venturesome for some tastes) at Ho Ho, Hing Loon, Sea Garden, Honey Court, and Chau's, all in or near the International District.
Lots of good Ethiopian food in the Central District.