Agree with OkieDave and prefer cheap campgrounds whenever possible. Simply cannot sleep in Walmart parking lots or rest areas; too noisy, too much light. State and national parks, and Forest Service campgrounds are usually a great deal. Private campgrounds usually offer a lot of services we never use or appreciate, catering more to families and longer stays, are priced accordingly, but are conveniently located close to highway.
Have a wonderful trip! Have been in Williams in October when it snowed, so be forewarned. Didn't stick and the Grand Canyon was beautiful the next day, but just sayin... If you weren't planning to go to the Canyon, please reconsider. You will be too close to pass on by.