I didn't read all the replies but Rock Chuck got it mostly correct. Get off I-84 at Brigham City & go through Logan, Utah, then Preston, stay to the right to Soda Springs & down the highway 34 down the Tin Cup highway into Freedom, Wyoming. You will drive right past Freedom Arms. Then you can go into Jackson Hole & on into the park.
Or you can drive around Palisades Reservoir & into Idaho Fall & back onto I-84 & go into West Yellowstone through Ashton, lots of great scenery either way. Great view of the Teton's from either side.
When you go into the Park be sure & take food & water, I don' think any restaurants are open in the park yet.

Dick