There are many safe ways to travel to Yellowstone with an RV. It just depends what you want to see on the way. From Louisiana you could head in the direction of I-40 west and tour the Grand Canyon in Arizona. Then you could take 89 north from Flagstaff, AZ and tour the awesome southern Utah national parks - Zion, Bryce, Capitol Reef or another direction to Monument Valley and Arches Nat'l Park. Then work your way up to Jackson, WY and stay at Colter Bay RV in Teton Nat'l Park. Then continue driving north through Yellowstone and exit to West Yellowstone if that's where you're planning to stay.
When finished with Yellowstone you could head back via the Dakotas and stop at other main spots. Lots to see in the West! Have fun planning.