We like the North enterance from Gardiner. It is through the historic Roosevelt (T.) Gate. Gardiner is always worth looking at the tourist "craziness" and Livingston at the I-90 exit gives you a chance to stock up at reasonable prices. This year will be out fifth and sixth visits.
Tom, Cheryl & Blossom(coonhound mix)
'05 Winnebago Journey 36G w/Cat. C-7 350 hp Freightliner XC
AFE air filter, aero turbine muffler, 4 FSD Konis, ultra track bell crank and Safe-T-Plus
I agree that most people advise coming in from the North due to taking I-90 over the Big Horn mountains. If you are unsure about mountain driving I would aviod Hwy 14 from Sheridan, WY over Burgess Junction, that is a pretty good pull. I would take Hwy 16 to Ten Sleep and then northwest to Cody and in the East entrance to Yellowstone. Very good road over Hwy 16 with passing lanes and pullouts. The East entrance is a good road as well. The construction in which they have been doing over the past several years has really paid off.
We have taken both Hwy 14/14A in all of our coaches as well as the Northeast entrance over the Beartooth Highway and would recommend it for scenery but if you are not comfortable or familiar with mountain driving I would avoid. Last year we took Hwy 16 to Cody because we had some new RVers traveling with us and they did not feel comfortable taking the steeper mountain passes.