It is easy to spend a fun and exciting day in Seattle without a car.  There are lots of ways to modify this itinerary. Add a ferry ride from the waterfront. Spending the day at the Seattle Center or at the aquarium would easily extend this from one day to several. Doing the underground tour in Pioneer Square or going to the Klondike Gold Rush Museum is fun and free. Don't forget the free buses downtown from the International District thru the main shopping areas of downtown.  In Summer you can also take the Water Taxi to West Seattle.

Here's an itinerary for one day: Start at Space Needle by taking monorail etc. Look around the Seattle Center or go to the Space Needle. If you're interested in music the EMP (Experience music project) is located at the foot of the space needle.  It has a lot of rock n roll memorabilia you may find interesting.  Take monorail back downtown.  You will be at 5th and Pine. Head west to the Pike Place Market. Look around and head for the Pike Place hill climb which is a series of steps that walk down to the waterfront. You will come out at the aquarium. Lots of stuff on the water, boat cruises, seafood, tourist traps etc. You can walk or take the streetcar to Pioneer Square. Do the underground tour and make sure to go to the Klondike Gold Rush Museum (free), especially if you are going to Alaska. When you get tired take a free bus back towards downtown. You can get an all day bus pass on the weekend and a visitor pass during the week. Even if you pay for the bus it won't be much. This itinerary will let you see lots of Seattle in a short amount of time.  For tour companies, try  Customized Tours toursofseattle.com.   They have been around forever and are the only ones that do a winery and waterfall tour.  It's nice cause it gets you outside the city.  Their city tour stops 4 times to let you get out, take pictures, and explore a little vs. just sitting on a bus.  Kids and families tend to like the Duck Tour that leaves from the Space Needle, but if you're serious about sightseeing the other tour companies tend to offer a little more comprehensive view of Seattle.

Grab "The Stranger" to find out what is going on in Seattle - free local paper released every thursday, available everywhere. It will be full of info on bars, music, moviehouses, and restaurants broken down by neighborhood. Beware that it is an alternative paper and you might not want the kids to read it.

There is a free waterfront trolley 'bus' that will take you from pioneer square all the way to the end of the waterfront where you can check out the new Sculpture Garden.  It's north of Pier 70. 

You can also take a day trip by bus from Seattle to the Boeing Factory, Snoqualmie Falls, or Mount Rainier.  Those will take quite a bit of time or the whole day.

One of the best values in Seattle sightseeing is the Seattle City Pass ©

Off-Site Links:

Seattle Tours
http://www.nps.gov/klse/index.htm
http://www.toursofseattle.com/
http://www.showmeseattle.com/