I've never ridden there, but I've driven through, maybe someone else has a better perspective, but from the drive, I would not say it's a very good cycling destination. It's really exposed to the elements, the scenic stuff is pretty far apart, and roads are gently rolling, no big climbs as I recall (from the car).
Also, it can be crazy-windy there. We had to pull over and wait on the side of the road for a couple of hours because we couldn't drive in our van, it was so windy. I would be prepared to alter your plans if there is high wind in the forecast - it's can be dangerous, and it whips up so much sand you can't really see the scenery anyway.
Sorry to be a downer, hope someone chimes in to correct me, but that was what I saw.
For riding around Monument Valley, I bet the bike shops in Moab would have the best info, I would call one of them. I have some road rides to recommend in the Moab area -Arches National Park, Canyonlans (both the north and south parts, and a loop from Monticello over the Abajos into Canyonlands is on my things to do list.
In Arizona, check out Hwy 191 between Alpine and Morency -92 miles, 13000' climbing, no services - spectacular. Also, Mt Graham as an out-and-back from Safford is a great ride. I'll be out there around the same time frame as you.