Go out the west entrance to Yellowstone, get on US-20 to Idaho Falls. From there pick up I-15 through SLC. Just south of Provo, UT at about exit 257 US-6/US-89 branches SE to Price, UT and down to I-70 near Green River, UT. Just east of Green River pick up US-191 to Moab. About 550 miles.
US-6 from I-15 to Price has a 30 mile climb to Soldier Summit on a very good 2 lane hwy with many long passing lanes on the uphill side. The grade ranges from 3% to a max of about 5%. From the summit you drop down to Price, again the grade is about 5%-6%, but it has some 40 and 50mph turns. On the down hill side you will want to manually down shift to 3rd gear and let the engine do the vast majority of the braking. Don't ride your brakes. When you have to use your brakes, press hard and drop your speed by at least 10-15mph in 10-15 seconds and then let the speed slowly build up using the engine braking. Remember, you can't go to slow going down hill. Don't worry about the traffic behind you.
Note that if you were to take I-15 to I-70 then east on I-70 to Green River, there are some pretty decent hills to go up and down. Then on I-70 about 40-50 miles west of Green River you have a pretty steep about 7% down grade for about 5 miles. That is dangerous, because with the nice wide interstate you will want to charge down that hill at 60-65mph and that makes it really difficult to slow the rig down. It can overheat your brakes, which makes them fade and loose their braking ability. Again slow and using engine braking is best.
I would suggest making your first days drive from West Yellowstone to Provo and spend the night in the Provo area (about 370 miles). Most all that is interstate. The second day, drive over the summit and on to Moab.
