OK -- you like the outdoors, but you have limited time. As someone who has spent a lot of recreational time in the Northwest, here is what I would do if I had only a couple of weeks -- I'd fly into Seattle or Bellingham and spend much of my time in the North Cascades. Our favorite area is east of Bellingham, east of the town of Glacier, in the Baker Snoqualmie National Forest (which would be a national park in any other state). It looks like this:
There is a huge variety of great hiking and camping in the area, very conveniently located. The only problems are (1) there are people on the trails -- this is not empty wilderness, and (2) depending on the year, snow can linger into August -- it is a good idea to bring microspikes, which are slip-on crampons. They work great, like 4 wheel drive for your boots!
Here is a link to our most recent posting about that area -- lots of photos:
Mt. Baker and Mt. Shuksan area, east of BellinghamThe posting before that one is about the North Cascades National Park, just south and east of Mt. Baker -- also a great area, great hiking, jaw-dropping scenery. The posting after that one is about the Olympics, which would certainly merit several days' worth of activity.
Wherever you go in Washington, bring good rain gear -- a high quality poncho, a goretex windbreaker, maybe even goretex wind pants, goretex boots, goretex underpants . . . . you get the idea. It does not rain all the time in the summer, but it can rain.
Since you have been to Banff, I won't drone on about how great that area is. If you'd like to see some photos, check out our blog posts for September and October of 2012. I hope this is helpful!