I have made alot of trips to West Yellowstone over the years and our groups preferred time to go is in January and February, but seems like we usually end up there in March. This is because we normally head to Revelstoke, B.C. in January and go to West or Cooke City as a second trip.
Depending on the type of riding you want to do will really be the big factor in timing. If you are just going to cruise the park, anytime Dec - March would work. If you want powder and back country riding Jan - Feb is the best.
I recommend staying at the Brandin' Iron Inn. The rates are reasonable, have 2 hot tubs and are within walking distance to restaurants, shopping and bars.
Our favorite haunts are the Happy Hour bar and Eino's. We have had some good food @ Alice's by Lions Head too.
If you are riding back country, make sure everyone in your group has a beacon, probe and shovel. And make sure everyone knows how to use them. If you are novice riders, a guide wouldn't hurt either. Avalanches in the mountains are no joke. Over the years our group has triggered about half a dozen avy's, but luckily none of us have been caught.
If you plan on trail riding, west has some of the most scenic trails around. Make sure to check the grooming schedules every morning as the trails can get really beat up over peak times.