Busskipper is correct about all of the SPs in this area and Hwy 12.
OK, campgrounds:
For the N Rim I like the USFS CG at Jacob Lake. Drive the car to the rim leaving just after sunset and count the mule deer... 200+ The Cafe/store there is not a rip off in the middle of nowhere.
For Bryce USFS Red Canyons on 12 after you get off of 89. Day trip back to Zion and do bottom half of 12 in the car.
From Bryce I take 22 straight N from it's entrance to Otter Creek SP CG for FHU.
When in doubt I head to a lake and Fish Lake has several USFS CGs before you get to I-70.
N of Moab on 128 there are a bunch of BLM sites on the CO river as Dennis suggested.
Not much until Flaming Gorge where there are plenty of USFS CGs... Firefighter's Memorial CG is nice.
I like 287 into Moran/Tetons and while I've gone the way that follows, in a couple places 28 is a little narrow if you meet another big rig but still doable... 191N from Rock Springs to Farson/28 to 287/Lander. The Sinks Canyon SP & CG there is very interesting and then 287 on to the Tetons is spectacular. 8 miles before you get to Moran is the Hatchet USFS CG and next to it is the Hatchet Cafe/Resort. Eat lunch there and get their WiFi code which I could access from the CG.
Instead of paying $60+ in W Yellowstone, three miles N is Baker's Hole USFS CG with a dump station and water... no reservations so show up early and you can probably get one of the 20 electric sites the first day and if not, the second day move to one for $20 with the old farts pass.
Hope this helps AND they are not full when I get there.:W