That trip out west sounds like one I took many decades ago...before the Internet and the GPS.
Back then the planning took one full year. To stay at National Parks you should have reservations and most must be made a year in advance. Back then all reservations were made by phone and U.S. Mail with "TW Services" an agency appointed by the Park Service. If you plan on staying at National Park Campgrounds plan a year ahead.
When we visited the Grand Canyon we camped in Williams, Arizona at a private campground. When at Yellowstone we camped for a week at Fishing Bridge A full hookup campground inside the National Park...it took a full week to cover that Park.
When we were touring both Bryce Canyon and Zion Canyon we stayed at nearby private campgrounds.
During our epic trip that lasted 6 weeks with our two children we toured almost all the National Parks in the West plus the 4 corners, Pikes Peak, the Indian cave ruins at Mesa Verde...we tried to cover it all. Because of advanced planning we allowed plenty of time for both exploring and travel. Durning the trip we limited travel to about 6 hours drive time per day. An over the road truck driver told me to figure on a maximum of 50 MPH for my calculations. That allowed for rest stops, lunch and gas stops etc. I suggest you use that target.
The longest travel day we scheduled was from Rapid City (Mt. Rushmore) thru the east end of Montana...that was 12+ long hours of empty country...eastern half of Montana was a very empty place back in the 1980s.
Our basic road plan was supplied by the AAA Auto Club....we used campground directories from Good Sam to locate campgrounds...we pre-reserved every night just to avoid surprises. Today the Internet would make the planning process a lot easier...back in 1983 it was a lot harder...the planning can be half the fun.
Advanced planning will make the trip more enjoyable.
During our epic trip we burned thru 2 trailer tires
(dessert heat not good).... We maintained a log of our trip but for some unknown reason never recorded gasoline costs...campground costs...yes...can you believe $6 a night for a private campground !....We took a million still pictures and using a newly invented camcorder created an epic two hour recording of the Adventure.
Plan ahead...enjoy the trip...see all you can...the Grand Canyon is fine but don't miss Bryce and Zion canyons and a lot of other sights along the way!
Happy Camping !!!