Pike Place Market. Its a complete tourist trap. This is also where the fish mongers who do the fish tossing are. Get there early to beat the crowds and take your time wandering around. The market has multiple buildings and most people only manage to really see the one. Wander through Post Alley (bubblegum wall) while you’re in that area as well.
Working your North from Pioneer square, you could stop at the main branch of the Seattle Public Library. If you like architecture, this building is for you. If you like libraries, this building is for you. Its pretty much all around win.
Woodland Park Zoo and the Fremont Troll. The zoo is just an amusing place. Its a fair size, though not huge. And it has an amazing penguin exhibit which everyone should see. Down the hill from it is the Fremont Troll (as seen in the movie 10 Things I Hate About You.) If you start early, Mae’s Phinney Ridge Cafe isn’t far from the zoo and does a pretty decent breakfast. Start there, do the zoo, wander down the hill and see the troll and then check out the grinder shop (Royal Grinders) near the statue of Lenin in Fremont.
Spend some time in the Pioneer Square area seeing some combination of King Street Station after the restoration is finished, the Underground Tour, the Gold Rush Museum, and Waterfall Garden Park. These are easy to do and a stones throw from the Mariners and SeaChickens/Sounders stadium if you want to try and be in the area
Go to a sporting event. Something like a Seattle Sounders soccer game. Or see the Sea Chickens (Seahawks) play American Throwball. Its easier to get tickets to see Rat City Rollerderby or the Seattle Storm (WNBA) though. Not necessarily easy to do, but an amazing experience.
Kerry Park is the iconic downtown Seattle skyline photo. Bhy Kracke Park just around the hill from it is a great view of Lake Union, Cap Hill and Downtown. Both parks are on Queen Anne Hill. They’re about 10 blocks apart. Both give great views of the city. I much prefer the Bhy Kracke Park view, but its worth seeing both. If you want a contrasting view, take the ferry from Seattle to Bainbridge Island and watch the skyline from across the bay.