You don't want to take the northeast entrance leaving the Park, since that goes over either the Beartooth Highway or the Chief Joseph Highway. Absolutely beautiful drives, but both definitely mountain driving.
Absolutely for sure follow the above advice!!
The only "flat" entrance into the park is the west one. South isn't bad (the one from the Tetons) - there's a climb, and as already pointed out, curvy roads exist everywhere in the area, including throughout Yellowstone, so you can't avoid them. East entrance is pretty winding, but take it easy (as someone already said) and the driver should avoid the temptation to watch the scenery. IMO, the 2 "worst" entrances to Yellowstone for those who want to avoid mountain driving are north (several hairpin turns) and for sure absolutely northeast. Speed limits inside the park top out at 45 mph, and there are frequent stops for wildlife & scenery viewing, photo ops and crazy fellow drivers who stop in the middle of the road. :) No rushing around going on in the park, that's for sure. By the time you've been there a few days and are leaving, you'll have the hang of it.
