The worse driving will be on I-80 - lots of trucks!
I prefer I-80 to Rawlins, WY then 287/26 to Moran and into Grand Teton NP. You'll cross the continental divide at Togwotee Pass. Westbound is 6% downhill for 17 miles but not steady downhill; mostly rolling hills. It is a gorgeous drive especially when you can see the mountains in the distance. - 1300 miles
The town of Lander has some nice things to explore if you're ready to spend a night before entering the park.
Find a good grocery before entering the park otherwise you'd have to drive into Jackson for groceries. You'd probably tour Jackson one day anyway but perhaps not right away. The grocery stores are just south of town - one on the left & one one the right. The stores are the best place for fuel, also.
The other route would be I-80/191 at Rock Springs to Jackson and into the park - 1360 miles. This is not particularly a pretty drive and you have to get through the madhouse of the town of Jackson with your RV. Tour Jackson without your RV.
Enjoy the park!! Awesome hiking! Take a float trip down the Snake River and you just might see a moose in the water with you. We did.
Study the park's website for lots of good information:
https://www.nps.gov/grte/index.htm