beerag04 said:
We did this exact same thing last year the first week of January, skied Big Sky and took one day to go into Yellowstone. We loved it. Highly recommend.
Our kids were not old enough for the snowmobiles, so we took one of the busses. Saw lots of bison, a few eagles, coyote, swans. You can watch old faithful and walk around the pools with just a handful of people, instead of the massive summer crowds. One of my favorite trips ever. Only catch is weather. We had a calm cloudy day with high in the teens, so it was great.
You can work around the crowds in the summer.
I'm an early riser. When I saw Old Faithful, there was only one other person watching. I didn't know he was there until it was done and I saw him walking away with his camera equipment.
In the summer months, anything after 10 am to 3 pm is horrible. Before that and the park is very manageable. After 3 to 5 pm, the place is a ghost town. Many of the visitors are leaving the park to go back to their hotels.
I would love to go to Yellowstone in the winter. I would absolutely do it if I were in your place.
Here's a shot of Old Faithful viewed only by me and one other person.