If you like good food: Go to the Chena Pump House Restaurant, AND to Pike's Landing. Both are fantastic.
If you like seeing horns and hides: Locate the Fairbanks Fur Tannery and go see what Allen has gotten in.
If you like local culture and fun: Go to Pioneer Park and check it out.
If you like scenic drives with a great destination: Drive up to Chena Hot Springs and spend the day. Have dinner, too. Beautiful!
If you have time: Denali Park is about 2-3 hours away. You can do a drive through in one day easily.
If you like smoked salmon: Locate Interior Fish Processors (aka Santa's Smokehouse) and buy a variety of their salmon offerings.
If you like custom knives and cool handwork: Locate The Cutting Edge in Fairbanks. Definitely worth seeing.
Have fun!