If best route is closest without much in the way of grades, I believe I'd go I-25 to Cheyenne, I-80 to Rawlins, US 287 to Moran (Tetons); US 89/287 to West Yellowstone (through Yellows Park). Slow traveling in Yellowstone due to speed limits, nothing in the way of extreme grades. IMHO.
594 mi, 10 hours 29 mins.
Get to see some of the Park before setting up in W. Yellowstone.
In Yellowstone, the only grade I can remember would be from the turn up near Grand Village to get up in the higher country on the way to Old Faithful. Can't think of any after that.
The is one pass on US US 287/26 to get to Moran, but can't recall it being much to worry about.
The Denver to Rock Springs to Idaho Falls Route to W. Yellowstone: 718 mi, 11 hours 34 mins.
The Colter Bay Visitor Center and Indian ArtsMuseum by the Colter Bay Campground in the Grand Tetons is really a neat one.
Post Edit: By entering via Moran, one does not have to deal with the traffic in Jackson's narrow streets & tight turns.
:CDesertHawk- Las Cruces, NM USA
*2015 Lance 1985~Casita de Campo~23' 4" Tongue to Bumper, Dinette Slide
160 watt Solar Panel/GoPower! Solar Controller
*2009 White Ford F-150 Reg. Cab
Long Bed with A.R.E. Molded Fiberglass Topper
*Previously~ 2005 16' Scamp