Regardless of the route, you're going to hit slopes on any of the major interstates heading east. If it happens, just slow down, downshift, uses your hazard flashers or strobe and take it easy. Believe me, you will not be the only group encountering the hills and you won't be the only slower vehicles on hills.
As for places to stay, just do your standard research and make sure you know if they are open if planning to wiz through and stop when needed. In parts of AZ, those higher elevation campgrounds may stop operating after October 1.
The only issue you will have is weather. Last week it snowed in northern AZ and as you drove I-40 through Flagstaff, you were greeted with snow cap mountains. The weird weather patterns are making themselves known right now so heavy rain could pummel you for a brief period.
Otherwise, enjoy the view.