Hey there,
I used to live in Fairbanks; well, Eielson AFB to be exact. Your best bet is to go to Mt. Auroa Ski Land. They have rentals and instruction. To get there take the Stese Highway out of town and at mile post 21 the turn off to Mt. Auroa is on the right. This is also an incredible place to watch and photograph the northern lights to be away from the light polution of Fairbanks.
Mt. Aurora Ski Land
Another ski area is at Moose Mountain out by the U of A Fairbanks. This is an interesting set up for they have no chairlifts;they bu people bck to the top of the hill. Moose Mountain also has a ski & snowboard school. Take Sheep Creek road out to Murphy Dome road, then turn off on moose Mountain road.
Shredthemoose
Once you get proficient at snowboarding, there are a lot of backcountry opportunities around Fairbanks; the best hills are out on the Chena Hot Springs road. A great area is the Granite Tors Trail. If the snow is good, there are huge, wide open glades up top. Down south on the Richardson Highway at paxon, you have the Alaska Range which as you know is huge.