Ok, a few notes...
Grand Canyon, Great Idea. It is spectacular.
Back way to Crown King. If this is a rental, check to see if it has "Arizona Pinstriping" if it does, you can do this, if not and the paint looks good, you may get dinged for striping their paint. It is also a long day. Also, if you don't do a lot of 4 Wheeling, skip ALL the obstacles.
Broken Arrow, Sedona. This is the BEST trail in the state. It is short, but it is a ton of fun. From a quick climb to Submarine Rock, to Chicken Point to the playground, there is just more and more rock crawling fun.
There are numerous other trails in Sedona, Oak Creek Homestead leads down to the creek and is a great spot to hang out
Greasy Spoon is just a really rough and rocky road, same as Schnebley Hill, but at least you have views on Schnebley.
There are TONS of fireroads all over the place, the Mogollon Rim is a spectacular drive over above Payson on road 303 Rimroad. There are even a few lakes up there. Fantastic views from the cliffs.
Tonto Natural Arch for when you feel like hiking down and see a really cool natural cave.
Many many more... Table Mesa, Montezuma's well, Honank and Palatki Ruins. Two guns and the Apache Death Cave.
This is all stuff in the northern part of the state. Heck, come to Prescott, a great little town.
If you want, I have run every one of these, multiple times. I can show you the routes.
Edited to add, there are trails up by Flagstaff as well as the Lava Tube, a natural cavern you can go hike in. There is also the Original Grand Canyon Airport, and abandoned airport near the canyon. Also Peach SPrings, there is the only road down into the GC and you can park your rig right at the water line of the Colorado River.
Freaking TONS of things to see
Like by Ash Fork, an old crumbling section of the original route 66, a steel dam and Johnson's tunnel, an old railway tunnel.
Second Edit. If you are going to be by Page, AZ, you can also see Horseshoe Bend and you have to go all the way out to Alstrom Point and Lone Rock. Alstrom for the cliffs and Lone rock for the beach.
Holy crap, lots of stuff I keep thinking about.