You can look up the famous places in Seattle. Some lesser known places:

• Klondike Gold Rush National Park in Pioneer Square. It is a museum staffed by park rangers, free, and the rangers sometimes give one-hour walking tours of the architecture of Pioneer Square.
• The Panama Hotel features in the book, Hotel at the Corner of Bitter and Sweet. There is a window in the tea room floor where you can see a box of possessions left by one of the Japanese-American families before they were interned during WWII.
• Water tower in Volunteer Park. Nice panoramic view of Seattle and Lake Washington.
• Pioneer Underground as mentioned in another answer.
• Seattle Public Library at 1000 4th Avenue. The stacks are a rising spiral hall that goes up 4–6 stories. Go to the top, to the corner near one of the elevators, and you can look down to the lobby. Wonderful architecture. The drinking chocolate at the espresso stand on the 4th Avenue side is also excellent.
• Ferry to/from West Seattle. Make sure it is running as service is intermittent, but scheduled.
• Ferry to/from Bainbridge Island and Bainbridge Island. The Bloedel Reserve is a peaceful walk, but pricey to get to if buses are not running.
• Museum of Glass in Tacoma and the Chihuly Glass Bridge. Famous is the Chihuly museum in Seattle Center.

Less known places in Seattle yet worth a Visit are:

1. Olympic Sculpture Park - It has a beach in there, along with beautiful park.
2. Japanese Garden - It is really wonderful garden and has a very minimal entry fees.
3. Washington Park Arboretum - It is great for hiking/jogging and you will love the ample amount of trees & plants in there.
4. Woodland Zoo
5. Seattle Aquarium
6. Underground Tour
7. Argosy Ferry & Other Ferries from Seattle Downtown Waterfront
8. Seattle Great Wheel near Downtown Waterfront
9. Gum Wall near Pike Place Market
10. Sky View Observatory - Columbia Center
11. Freeway Park
12. “Veggie Grill” & “Plum Bistro” - Vegan Restaurants
13. “Frankie & Jo’s” & “Sugar Plum” - Vegan Ice-Cream Shop