Asahikawa is still relatively unexplored by ski tourists and and there are few International ski schools like you would find in more popular areas like Hakuba or Niseko. Your best bet for English speaking ski and snowboard lessons is with Miura Snow Sports at Kamui Ski Links. They run kids lessons, private lessons and off piste workshops.
Asahikawa is great as a base for riding the Daisetsuzan ranges. To get the most of your trip, engage with experienced guiding companies that know these mountains best.
Mountain Flow guiding company specializes in backcountry tours to the Daisetsuzan ranges.
Trive Mountain Guides run by Kazuyoshi Arai have a Asahikawa Backcountry Tour focussed on Asahikawa, Furano and Horokanai.
Tomamu based Alpine Backcountry Guides can customise a backcountry tour to Asahidake and the Tokachi mountains.
Tengu Cat Ski Guides run by Naoyuki Kato runs cat ski tours to the now defunct Kitataisetsu Ski Area.