I used to work in Tokyo and several winter weekends went off skiing. I don't know what level of skiing you are interested in. We were limited to weekends and drove to our destinations.
I'd suggest Naeba, it has a whole range of runs. You can get there by train from Tokyo, and there are other westerners so the locals won't run away if you try and speak to them lol. You'll find all the info you need here...
http://www.snowjapan.com/e/spotlight/mount-nae ba.html
After a hard day skiing it's really nice to relax and scrub-up in an onsen (hot spring), though I'm not sure if they have mixed sex one. Then there is a traditional type of restaurant always owned by ex-sumo wrestlers, where they serve a kind of shabu-shabu from a big communal steel 'wok' on a burner in front of you. It is more like a stew. Great fun, you can barely walk afterwards.
A couple of beers in the Prince hotel at night is also nice, it is at the base of the main Naeba runs, and the hotel has a huge glass wall looking up the mountain, so you can watch the night skiiing.
When I was there mid-90's it was still hard to get info (travel, passes etc) in English locally. The above website looks good, I'd do all your research before you go.
Have fun, we certainly did!
p.s. The other place we used to go was Yuzawa Park, much smaller, lower key, no other westerners, still great though.