Welcome to the forums...it would help if we knew what you are driving, you can even put that info in your profile. I would suggest a slight different route than Busskipper...at Rawlins take 287 to Moran Junction/Tetons. Yes, you go over a fairly high pass put the climb up to it is a easy straight pull on a very good road. The view of the Tetons is spectacular and by manually gearing down the steeper decent is no big deal. This will give more coincidence in tackling mountains. 287 will take you into Yellowstone and all the way to West Yellowstone by way of Old Faithful.
I always recommend a stop in Lander to see Sinks Canyon SP...excellent small visitor center and a river going underground and coming back up a 1/2 mile away is something your children will not see anywhere else.
