#1 advice from me, try not to see so much that you really do not see much of anything. That's allot of ground to cover, don't spend it all in the car. Here are some Day hikes I have done that I would highly recommend:
Lake Louis (I stayed in the hostel, I am cheap but too lazy to camp, join ACC will allow you to do this)-
Beehives (easy) 1/2 day.
Mt Fairview - (moderate 1 day)
Jasper - Mt Wilcox (difficult scramble), my group could not find the way up to the summit without what appeared to be unroped rock climbing to us, but it is said to be doable without rock climbing. Fabulous views of Columbia icefields and Athabasca Mt & glacier, even as high as we got. (1 day for RT hike, 1 day RT travel from trans canada highway).
North Cascades -
Hidden Lakes Lookout & Peak, (moderate 1 day) possibly my favorite hike anywhere for views
Sourdough Mt (moderate 1 day) allot of vertical, 5200', but very good foot friendly trail so not too hard. Gorgeous summit views, famous beatnick hangout
Hannegan Peak (moderate 1 day) - great views of Baker, Shuksan and northern pickets. Also do drive to the world famous view of Mt Shuksan from a lake with a huge parking lot, but well worth the drive. This is the view on the NP monopoly game box cover (which incidently is not actually in a NP). Do in same day as Hannegan peak climb. Near Candian border.
Mt Rainier NP -
Tatoosh range plummer peak (easy, 1/2 day), great views of Rainier (ofcourse from south). Also you can try neighboring Castle peak, which you are supposed to be able to scramble up, but I could not find the route that did not appear vertical rock climbing to me, like Wilcox.
Shriner peak - ( moderate 1 day ) , great views of Rainer upper mountain from the east, where you can pick out the entire Disspointment cleaver route on the mountain.
Camp Muir (already covered above)
Burroughs Mt & sunrise (easy half day)
Southern WA -
Mt. St Helen's - Unique hike, do it at least once in your life if you get the chance.
As stated above, check with Rangers ahead of time shortly before you head out there to see what is passable after this winter's storms out there.