Presuming Flying in to Salt Lake City, here's a rough circle trip.
1) arrive SLC
2) Drive to Price - Nine Mile Canyon for Petroglyphs and desert scenics
3) Moab -
Arches
Drive Up 128 to Castle Valley
Canyonlands North area - Island in the Sky; Rent 4WD in Moab for White Rim Trail; Deadhorse Point
Along River West of Moab - drive north and south (Petroglyhs, river, inquire about directions to birthing petroglyph)
South of Moab - Canyonlands Needles District, and along highway 128 - newspaper rock, side trip to Hart Point. Get a permit in advance,rent 4WD and drive back roads in Needle district.
Nearby - the remaining part of Canyonlands called the Maze (4WD territory) plus a hike down to "The Great Panel". Inquire at the visitor centers for directions.
When you are exhausted at Moab, leave down and head south...
Reaching Monticello -- consider a side trip into SW Colorado, or just head south to Bluff . If Colorado go to Dolores-then up the canyon thru Rico to Lizard Head Pass and down to Telluride. You can take this a bit further - drive through Montrose to Ouray, up Million Dollar Highway and Down through Silverton to Durango. Near Durango is Mesa Verde. All depends upon whether you have 2,3,4 weeks on your trip? Lots to see still back in Utah
After Mesa Verde - decision time....Chaco Canyon or back to Utah? Presuming Utah -
Head west (maybe a stop at Canyons of the Ancients in UTE Tribal Park) until you reach BBluff. Talk with locals (try the Wild Rivers Bunch) about Butler Wash, and the Sand Creek petroglyphs. Lots of Anazazi ruins and petroglyphs as well as the Sandstone Bluffs along the San Juan.
Leaving Bluff - go to Mexican Hat. Stop At Goosenecks of the San Juan, and Muley Point (Its odd to have dirt state highway but that's what they have to climb the mesa).

At Mexican Hat you have to think about that side trip to Monument Valley....
Head down through Arizona to Kayenta then back up to Utah at Kanab.
Kerry's right - North Rim is snowed in, so unless you are going after late June this year, you'll need to rent a snowmobile....
But you can go to Toroweap Point...look for the long washboarded (but flat) dirt road out of Fredonia.
then on to Zion, Bryce and Cedar Breaks....but You've missed the Escalante....and the backcountry in the center of this trip.
There is at least a month worth of good travel, so you'll have to pick and choose. Google some of these names and figure out whether you want to choose
Have a great trip
Don