Natural wonders: Yosemite, the Grand Canyon and the Sonora Desert
The highlights of my trip were definitely when I got out of the cities and went hiking.
Stunningly beautiful and really peaceful and serene, Yosemite is hardly marred at all by the huge numbers of visitors. (If you climb up onto the rim and keep going a bit you can get away from the crowds anyway.) Four full days would probably be enough to do the nicest dayhikes from the valley floor. (I would have liked to go backcountry but wasn't game to do it on my own; one day I want to come back and walk the Pacific Rim Crest Trail, or at least bits of it.) If you are passing through California Yosemite should be on the must-see list - if nothing else there are day tours by coach from SF and LA.
The Grand Canyon
Should also be put on your must-see list. I walked down to the river and back, which is a pleasant day hike (if you are fit just ignore the signs telling you not to try doing this in one day). The view from the rim is just mind-blowing, and geologists will go gaga at all the exposed strata. Once again, you don't have to stay long to appreciate it - it's worth going there just for an hour on the rim gawking at the view.
The Sonoran Desert
I managed to fit in one day hike in the Superstition Mountains to the east of Phoenix. Just like in all those Westerns, except more so! Weirdly shaped rock formations, winding canyons and forests of Saguaro cacti.