I have used every entrance/exit to Yellowstone.
The most difficult exit would be the one over the Beartooths, I would not recommend it to you if you are not comfortable driving in the mountains.
The next possibility would also involve the NE exit at Cooke City but you would take the Chief Joseph Hwy. After you go over Dead Indian Pass you could head South towards Cody or North to I 90 at Laurel MT. This is defiantly a steep winding mountain rode but a good road and very scenic.
The next possibility would be the North entrance at Gardnier, head North to I 90 at Livingston MT. This route is relatively flat.
The next possibility would be the East entrance to Cody. Not a bad road at all, very scenic.
Any of the choices that take you to Cody will require you to get over or around the Big Horn mountains. Highway 16 is a better choice than Hwy 14 or 14A. Driving to Casper will lengthen the route but be flat.
Any choice that takes you to I 90 will lead you to the Black Hills, longer but not a bad drive.
I live in MN so have done this many times. The most common route for me to take is to use the East entrance to Cody and the 16 to Buffalo WY where you pick up I 90.
