Hmmm... Its a dodgy time of year for snow. I would say just to be safe, you should go to Nagano. Its not that far. Tsugaike Kogen has great learners slopes, as does Nozawa Onsen. Your family will get a great Japan experience if you take them to Nozawa, its a beautiful town.
Can anyone here think of a worse way to spend Christmas than at a ski resort in Hyogo with no snow on it?!
About the trains, if you dont want to have to change trains in Nagoya, I think there is one express train a day that goes all the way to Nagano from Osaka. From Matsumoto or Nagano city you will need to take