What do you like to do?
If it's general exploring, you could bus over to the University of Washington and explore campus. If it's a nice day, it's a nice walk through the center of campus. And if you are in the U-District you could walk around University Ave, and visit the Campus bookstore. It's one of the largest independent book store west of the Mississippi.
Along those lines, you could take a bus to the Fremont neighborhood. It's a very eclectic place....a troll under a bridge, a statue of Lenin, the "center of the universe." Lots of interesting shops.
Pike Place Market, and a walk along the waterfront. Ye Olde Curiosity Shop is not as crazy as it used to be, but still not your average tourist shop. Have lunch at Ivar's.
Woodland Park Zoo is an excellent zoo.
And if you like baseball, the Seattle Mariner's baseball stadium is a beautiful ball park. The Pyramid brewery's original location is across the street from the stadium.
And the last thing I thought of, was just ride one of the ferries from downtown Seattle, through the islands over to Bremerton and back.