Some destinations that come to mind:
- The Badlands (in South Dakota)
- Theodore Roosevelt National Park (North Dakota)
- The Grand Canyon (Arizona)
- The Painted Desert and Petrified Forest (Arizona)
- Central Arizona (between, say, Phoenix and Flagstaff)
Of course, as you've noticed, the Arizona locations don't tend to have grassland nearby. The Dakotas, however, have lots of both desert and grasslands, sometimes quite close together due to terrain. Driving around the Western areas of the Dakotas and Eastern areas of Wyoming and Montana, you will see lots of enormous ranches with large herds of cattle. Also, you're likely to see at least a few herds of buffalo roaming around in the grasslands, especially in the national or state parks.
One note on the Dakotas: you probably don't want to visit in the winter unless you really like snow and very cold temperatures (think -20 to -40 C.) While the temperatures there may be +40 C in the summer, they can get insanely cold in the winter. Many of the scenic roads there are completely closed during the entire winter season.