[twindman]

Been to Yellowstone probably 5 times in the last 10 years. Unless you are experienced stay away from 14A in Wyoming. My 'Mountainous Driving' book calls this road the 'Oh My God!' road. There are 8 miles at 8% grade and then just to let you know that isn't bad, there is then 10 miles of 10% grade. Better going up than down tho!!!
However, I would recommend seeing Devils Tower on the way, and then take 14 or 16 thru the Big Horns (My favorite place in the west) on the way to Cody. I think hiway 16 is Shell Canyon - a very pretty drive. Unfortunately the driver doesn't get to see all the scenery (ask my how I know) because you can't really take your eyes off the road too long out west.
As for the east Yellowstone entrance - I don't consider it challenging at all, but I DO like hills and curves. Yes the other 3 side have a little more level roads, but the east side should not be scary.
And be advised about the KOA (if my memory is right), it is kind of out of town. Seems like each park in Cody has something at least slightly wrong with it - none have everything I want.
But I love Wyoming and my cousin lives in Powell (20 miles from Cody).