Without a car it may be difficult to get around much. Sea-Tac airport is south of the main part of town. There is a Light Rail train that runs from the airport into Seattle, which is convenient and fairly inexpensive. Here is their website's schedule: http://www.soundtransit.org/Schedules/Central-Link-light-rail.xml
Once in downtown Seattle, there are a host of things to do. You don't mention what time of year you'll be in town, so the weather may have a bit of impact on what you decide to do. There are lots of shopping and sightseeing opportunities, (Pike Place Market, Space Needle, and so forth.)
If the weather is good, making your way to the ferry terminal could give you a mini-cruise on Puget Sound if you walked on a ferry to Bremerton or Bainbridge Island. Cost is only a few dollars per person for that.
If you'll be here between May 24th, 2012, and January 6th, 2013, the King Tut Exhibit will be in town at the Pacific Science Center (located at the base of the Space Needle.) You could get some serious museum time with that exhibit. I saw it in the 1970's when it was here last, and it was amazing. http://www.pacificsciencecenter.org/King-Tut/king-tut-the-exhibition
Others will certainly have their own ideas, but this can get you started.