Spring is the time to drive Mt. Rainier NP... perhaps during the week to avoid the crowds. The whole drive from Elbe (706) to Yakima (41) is a must. It's got it all, great twisties, ups and downs, views to die for and they retar and fix the roads over in that area so often you might be tempted to eat off them!
The whole state is a driver's paradise really. The southeastern portion is an isolated semi-desert with wide open views and good roads. The western shoreline is supposedly among the most beautiful in the country (if not the world) although I have not myself traveled it. And the central areas are sparsely populated farmland with varied topology. You can't go wrong anywhere.