Here are some personal recommendations of areas in Western Washington that hold a special place in my heart.
North Cascades Highway and Ross Lake National Recreation Area. Highway 20, the North Cascades Highway East from Rockport, WA into the mountains is incredible. Very scenic. Along the way, you will pass through the Skagit Valley hydroelectric project. This is a series of dams and other structures. All large, and all in rough terrain. I went up there every year for most of my life and I look in awe every time. Near the center of the Cascades on highway 20 you will enter the Ross Lake Recreation Area. Ross lake is formed by Ross Dam (part of the above mentioned hydro complex). Beautiful area. I used to go to the Ross Lake Resort for a week of fly fishing every year. Drive to Diablo Lake (formed by Diablo Dam, take the Puget Power tugboat up to the base of Ross Dam, load onto a flatbed truck and drive through the forest to the top of Ross Dam, then take a boat across the lake to the resort. Alternatively you can hike in from Highway 20. If you're in to hiking, canoeing, etc. The area can't be beat.
http://www.rosslakeresort.com and tell Tom I said hi (I'm the skinny kid who used to come up with Ralph R. every year).
Hurricane Ridge. I can't say enough about this area in the Olympic Mountains. It will blow you away.
Hoe Rain Forest area. What? A Rain Forest in Washington? You have to see it to believe it.
Pacific Beach. This is a small state park on the pacific coast. Another traditional camping spot for me. This is near Ocean City, Copalis, and Moclips. It has the added bonus of being just a bit North of the Cache Across America - Washington cache. Bring your kite. In many areas here you can drive right on the beach (it is a state highway). Heed the cautions if you do. Also just south of Pacific beach is the Sandpiper. It's a cool store full of all sorts of wild items. Worth a look. I spent many late nights with my father in Moclips. We'd walk the beach at night by lamp after a storm, looking for glass floats that made the journey from the far east. We have found several together. Probably a rare find these days.
Take a Ferry Ride. From the main dock to Bremerton and back is nice. Or, drive from Seattle, across the Tacoma Narrows Bridge, then North to Bremerton and take the ferry back to Seattle. If you have the time to take a Ferry up to the San Juan Islands, do it.
All these are long day trips from the Seattle area. If you like, I can make some recommendations for things to do in town.