I suspect you’ll be disappointed with Rt. 66. Not much there anymore. Make sure you find your way up into Utah and don’t miss Grand Staircase Escalante. Rt. 12 from Torrey, Utah down to Bryce Canyon. Spectacular ride! Stay at the Broken Spur Inn in Torrey. For a unique experience, you can stay in a covered wagon. Here’s a video of the ride.
If you’re coming up from I-40, make a stop at the North Rim of Grand Canyon. Less populated than the South Rim but very special. Either plan on camping or get advanced reservations.
Many of the National Park campgrounds are full early in the day. However, there’s usually good private campgrounds around the areas. You can camp anywhere on Bureau of Land Management (BLM) property. Be aware that there’s usually no camping on Indian Reservations and some of the reservations in the Dakotas can be really huge. I had to ride over 600 miles one day in South Dakota before I got out of the reservations and found a small motel.