Best route is probably I35 to I70 into Denver, then I25 north to I80 in Wyoming. Take that to Rawlins, then follow US287 all the way to Teton/Yellowstone.
Not sure how many days you plan for the drive each way, but if you're doing three days each way (roughly 500 miles a day), a great stop would be Loveland, CO, just north of Denver. Johnson's Corner RV park is nice and they have "world famous" cinnamon rolls. They actually are quite tasty. And about 10 miles north is Fort Collins - a great college town with a nice downtown area for walking/eating.
I'd also suggest staying at Fishing Bridge in Yellowstone. It's the only RV park inside Yellowstone, and has hookups. You're going to be driving A LOT around the park, and it's a nice, central location.