The route we take from the St Louis, MO area to Yellowstone is I-70, I-435, I-29, in Iowa we connect to Hwy 2 and take that to I-80 and exit to Hwy 287 near Rawlins, WY 287 to Hwy 26 and then hwy 191 north at Moran Junction. Hwy 191 takes you through part of the Grand Tetons and into the south entrance of Yellowstone.
BTW, regardless of which entrance you use the fee you pay to enter is good for both Yellowstone and Grand Tetons.
Back in Sep 2004 we were departing the Madison campground and heading to the east entrance/exit when it started to snowing lightly. I hate towing in snow.
That year the road to the east entrance was under reconstruction and it had been taken down to the hard-packed mud. That road is a steep grade and I was puckered up going down the long, narrow, two-lane, and wet hard-packed mud during a light snow. As we descended, Joyce could be heard alternating between saying prayers and screaming her “silent” scream. I still tease her about being able to hear her “silent” screams.
Headed down to the east entrance.
We made it to the bottom without incident but I pulled over at the first opportunity so I could pry my fingers from the steering wheel.