If you get the chance go and drink and dine at the Pump House on the Chena River. There is also Pike's Landing on airport side of the river (you'll pass it leaving the airport.) The University of Alaska Fairbanks has a great museum. You can also head out to the Trans-Alaska Pipeline and listen to it going dry. While headed towards the pipeline you can go gold panning at one of the gold dredges. I don't know whether it will still be on during your stay but Alaskaland has a home grow stage show. If you are in to old airplanes still flying you can go to the east ramp and smell the fabric and dope. Old planes don't die, they go to Alaska.
I second the previous responses, though the really hard drinking is usually done by the year-round residents.
Denali National Park is a good road trip (2-3 hrs) from there, if you're into that & can go that far. Should be late Autumn by the time you arrive; very pretty up there.
Drive out the Chena highway(?) towards Chena Hot Springs. The scenery is pretty cool and the hot springs used to be a great place to relax. They used to have a bike ride during the summer from Fairbanks to the hot springs (60 miles); was lots of fun, though I think that's done by the time you get there. The Granite Tors hike is along that highway; one of my favorites when I lived up there. It's a good day-hike with great scenery and views.