We have just come up Highway 12 from Bryce Canyon a week ago. Can't anything to what has already been suggested except for this one. If you want to stay in any of the Utah State Parks, you had better get reservations!! Kodachrome SP was full when we took a day trip there with the car from Bryce. Escalante Petrified SP was full when we pulled into there. We could have hung around for a dry camping spot, but it was still early in the day and we didn't feel the need to that. If they were full - this was the middle of the week in April!!- I can imagine what they will be like later!
Correct, campgrounds at Red Canyon, Bryce Canyon NP and Capital Reef NP Fruita campground (Not to be missed, one of our all time favorite national park campgrounds) are first come first serve.
Between Red Canyon and the entrance to Bryce NP, there is a GREAT wildlife museum. Be sure to visit that - it is one of the very best that we have visited anywhere!
Agree there are some great CGs that are first-come, no reservations around there.
But an update to other info posted...we stayed in Bryce Canyon last year for several nights and were able to reserve in advance online. (no hookups). The CG was full of international travelers in rental RVs in August. There are two CGs. Second one may be no reservations.
Here's the link to American By-Way12, this Highway is a wonderful route to take to see your first time of Utah. Having done this route several time your have little problems finding camping areas to stay in, I would suggest that you relocal your base camp for the different areas for you'll find your self doing more driving that seeing the sights. Truck Camper Trip Report (link below)has many reports with lots of information and photos of the very areas you want to see.
As far as getting fuel along this route it's of no problems. PM me if you have or need more detail information.