It's been awhile, but I believe the east entrance and the road leading up to it has some big curvy grades. Looking at the map, you could take I-70 and then work you're way north and come in through the south entrance. The south entrance takes you through Jackson Hole and past the Grand Tetons.
There is a nice campground, for the area, in Jackson Hole, called the Virginian. It fills up fast. From Jackson Hole you can see all of the Grand Tetons and surrounding area easily with a car.
We leave for Yellowstone (Fishing Bridge) in two weeks, will stay in Yellowstone for a week, three days in Jackson Hole and then onto Bryce/Zion for three days.
You don't say what time of year you're going, but if it's soon, the weather may play a factor coming in the east entrance.
Always check the Yellowstone road conditions because during the Spring and Summer they do all their road repairs and often roads (entrances) can be closed.