we've done most of this route many times in our rig.
from GC to Moab, take the east gate out of the park and take 160 to 191 and north to I 70. Then west to 191 N, also called 6, up Vernal and on north to 89 and up to Jackson WY etc. and into Yellowstone.
We love going this route.
If you want to see some of Monument Valley, turn north at Kayenta and go up to 191 through Mexican Hat. When you enter Mexican Hat it is down a sharp descent of about 3 blocks long with a 90% right turn at the bottom. No big deal, but it does make you think the first time. :)
Vernal UT has a walmart if you are needing one by then.
There is a really nice campground up at the top of the Flaming Gorge area called the Firefighters' Memorial campground. water and dump and some elec hookups.
On north, there is another walmart at Rock Springs. We usually make it past Pinedale, WY about 20 miles to the Green River where there is a federal campground right on the river, with water and dump and elec.
That puts us just right for arriving early at Jackson and we stay at the Gros Ventre campground, about 7 miles north and east of town..in the National park.. on the east side of the park.