TexasChaps
We made the trip you are planning just in reverse about three years ago (We traveled from Bryce N.P. south to Williams AZ). Our route was from Bryce (Highway 12 to 89). Highway 89 south to Kanab. We then headed East on 89 to Page then continued south on 89 to Flagstaff. At this point we turned west for Williams.
We have travelled East & West on Highway 40 and South on 17. I have no problem suggesting either of these route for you. I didn't see where in Texas your starting point was.
There were plenty of fuel stops and services along the way. It is desert and we were passing through in Sept. & Oct. In Kanab there is a great Western Movie Museum. Many (many) things to stop and see so take your time. There were no major hills.
Enjoy your trip.
Hap