Not knowing where you're leaving, I chose Lafayette, LA. From there take I-49 to Shreveport; I-20W to Dallas; I-35 N to I-70 at Saline, KS; I-70 W to Denver; North on I-25 to I-80West. See my above post on one option from I-80 at Rock Springs. This would bypass your driving through both Grand Teton & Yellowstone parks to get to W. Yellowstone if you took 287 off I-80. You'd go to Idaho Springs and then north faster than 287 but you'd add 100 miles. Going through the parks is o.k. but it's slow-going. Park your MH then do your driving through the parks.
You'll have 1800-2000 miles from Louisiana and it's all easy driving.
Ourselves, we dislike interstates and big cities so we would find secondary routes and enjoy the scenery.