Hi ekfeejay,If you are coming in from the west on I90, jump off at Columbus and take Rt 78 through Absarokee to Red Lodge. I went through there today and it is just gorgeous. An alternative is to go to Laurel and take Rt 212 but that is pretty boring. Went over the Pass today about 11 am, 52 degrees F with cool wind blowing. The place was rotten with harleys as there was a rally in Red Lodge. Cooke City is the NE armpit of Yellowstone. I stopped for a burger (not bad) but the water in the place was off. Some kind of general water supply failure in the town. As Wheaton says, there is a super 8 and a number of other places but it is all pretty shabby. There is a bunch of construction west of town, had a 30 minute wait to get through. Once into the park the road is pretty good as they have upgraded it in the past few years. The Lamar valley between the NE entrance and Tower Falls is pretty but nothing special. Not many animals around this time of year. Only a few buffalo and I saw one lonely cow elk coming out of Mammoth.
I'd suggest you skip Cooke City altogether and stay in Columbus or Red Lodge and then go over the Beartooth and then head south to Cody and then in the East entrance. That would (to me) be way more interesting. And the road from the Beartooth Pass to Cody, called the Chief Joseph Highway, is superb. Not to be missed.
Cheers and have a good trip.