I did a couple drive holidays over the last 5 years where I covered California, Nevada, Utah and Arizona. I visited the following National Parks over the two holidays : Death Valley, Sequoia, Yosemite, Joshua Tree, Grand Canyon, Monument Valley (not technically a National Park), Arches, Byrce Canyon and Zion Canyon.
I would recommend any of the above particularly Yosemite, Death Valley, Grand Canyon and Monument Valley.
The best way to get around if you can is by hiring a car as you are quite flexible; try to have at least two drivers as it will tire you out big time otherwise. As far as accommodation is concerned, try to avoid staying in the parks themsleves as they are much more expensive (however it can be a long drive to the nearest motel outside the park e.g. Yosemite). There are plenty of great roadside motels and they are almost always a good standard