Geaux,
Sounds like you'll be trying to cover a lot of territory. Just remember that the lower altitude terrain, like Monument Valley, Moab, Arches NP will likely be close to 100 degrees in June. You'll have lots to do in the higher altitude areas. Our snowpack is well below normal this year and you should be able to get up high. In a normal year it might be 4th of July before you'd get over some of those 4WD passes. I'll echo 7mm08fans reco for the Black Canyon of the Gunnison, it's one of my favorite places on earth, especially from the bottom looking up when the Salmon fly hatch is on.
