We have been to AZ and the GC several times...
Just before Amarillo, TX is the largest cross in North America. Just off I-40; you can spend the night free with 110 hookup.
In Flagstaff, I would take 89 north and see Wupaki NM and the east side of the GC...In the GC, you can park and take a shuttle bus to different locations. Better than trying to find a parking space.
Next, on the way out of the GC go south into Sedona, AZ and Cottowood...Several things to see.
Head down to Tuscon and i-10 home...you might as well see Tombstone, AZ...
Good traveling.