If you plan to stay in West Yellowstone, I would suggest going up thru Idaho - you can cut off from I-80 to I-15 on the I-84 spur route to Ogden UT, which is a bit less steep of a pull, or you could take US 30 from I-80 in sw Wyoming, which will connect with I-15 in southern Idaho (missing all the SLC traffic). Some decent CGs along the way: Montpelier KOA is nice; Pocatello KOA is a good overnight stop; decent little county-run CG near (but not right next to) I-15 in Shelley. Grizzly RV is a great CG, IMO; we almost always stay there when visiting the park. Should have room for your rig; not many CGs do for something that size, especially inside the park.
You can come down to West Yellowstone from the north (I-90), but will be driving through Bozeman traffic for a while. Take 191 south from Bozeman to West Yellowstone - very nice drive on a good 2-lane road.
Haven't been the Casper-Riverton route but it's probably fine; Dubois is a nice stop (we have been at the KOA twice and like it as well), and Togwotee Pass, east of Dubois, is a fairly easy pass with beautiful scenery, no steep climbs, good road. Of course, if you go in any other park entrance, you will be driving through the park to get to West - it's do-able, but a lot easier in just the toad than with your entire rig.
Definitely avoid the NE and East entrances - especially NE!