I spent many, many summers in Seattle as a kid. So I did all the typical touristy stuff, the space needle, etc. We even drove to Vancouver once (well worth it.)
I feel that not enough people who visit Seattle (or the state of Washington, for that matter) end up exploring the majesty that is the San Juan Islands. I didn't end up doing it until I was much, much older. It is absolutely gorgeous, feels worlds away from Seattle and is relatively easy to get to thanks to the state ferry system. You won't have to take some crazy expensive tour to get there.
You probably won't have time to go to Alaska, and might only be in town for a week, so go for the next best thing. Sea kayaking is really popular in the area and a number of outfits will rent you one. If you get lucky or time your visit right, you might even get to see a pod of orcas - one of the main attractions. In my mind, a must-see of Washington.
Google map: tinyurl.com/yks76fb