I've done it about 6 times...probably the best outdoor winter thing EVER.
You don't say if your doing a Snowcaoch or doing Snowmobiles. If your older...really old...snow coachs are just fine. Excellant way to see the Park. Nice and warm and comfy. I'd recomend if your doing the coach, stay in Jackson and book out of the South entrance. Jacksons just a cool town with lots to do...but, if your riding the "sleds", West Yellowstone at the West Entrance is the place to go, for two reasons..."The Snowmoblie Capitol of the World"...
1. You get up out of your bed, hit the sled and in less than 2 minutes you are in the Park. The park it's self is not the "greatness" it used to be.(I don't mean the sites...just the way they let you view them)
They have run off the old time Rangers and replaced them with Clinton era environuts who think the park is theirs and you have no right to be there. You have to have a Guide now and book through a certified park snowmobile company.(1st time there it's good idea to have a guide anyhow...and you don't have a choice anymore)
This new breed of Rangers are JERKS. They will follow you from dawn to dusk and actually ticket you and kick you out of the Park just for getting out of line of your group or falling too far behind your group. These guys have runined the park for the serious rider....but...that said...
Yellowstone in Winter is a site to behold and two days in the Park is a MUST. Personally, a trip down to the South and Old Faithful Lodge for lunch, maybe down futher and over to Yellowstone Falls, well, you can work that for a couple of days and see all the Gyser Basines you can stand. There are a few of the bigger ones off towards the North (this is from the West Entrance) and if you go far enough north towards Mamouth, there's some awesome stuff...but it's a hard ride and if you don't have a bit of experience on the sleds...well, anybody can drive them, but weatherwise, think motercycle at 50 miles an hour at minue 40 degrees...now, don't let that scare you, cause your having so much fun, that the cold is simply not really a bother after the first 30 minutes in the morning...and of course, there's always a "warming hut" just around the bend.
But, the second reason I'd chose West Yellowstone is that after yoyu've seen the Park you can spend a few days in the National forest on the South side of West Yellowstone...beautiful Alpine like views deep snow and you can OPEN THE SLED UP(many of my Pro-snowmobile driving friends say it's the best snow in the Rockies)...Do it like a trail ride, stop at taverns and bars along the way...an all around great time.
In short, I'd do both...a few days in the Park to see the "Wonder of the world", and a few days in the National Forest.
I've always had good luck booking through West Yellowstone Tour and Travel. They have good properites and good sleds...Three Bears Lodge is pretty nice...good rooms, nice little bar and a pretty darned good resturant.
Oh, and there's a Steak House outside of town...can't remember the name right off...locals will know it...you pick your meat, they cut it and then you cook it...a real Snowmobilers haunt...people come out of the woods on the sleds for a night of drinks and steaks...check it out. If I can think of the name I'll post it.
If you haven't book flights and such...let me know, and I'll give you what I think is the best way to get there and it's probably not what your travel agnet thought.