Pikes Place market is good to go to as a tourist, and fun to keep going to. They have (I think they still have) a head shop. Hey, you can buy a magic mushroom kit, take it back to where you're from, and grow a cash crop in your dark basement. I digress...
Prepare to get lost. And bad. Take a Ferry ride, outside, not inside. It does some cool shit to your hair. Go see the mountains, and get out of Seattle if you have the chance, and see the mountains like Lillitu suggested.
Walk, then walk some more. Drive, get lost, drive some more. Hit some bars, ask for directions, get lost again. Prepare yourself now! You will get lost. And its fucking fun, because you see way more stuff that way.