Call Denali Nat'l Park before you head down there. Generally speaking, the road isn't open very far at all in the winter. In the summer you can only drive the first 16 miles of the road into the park; if you want to go further you need to get a bus ticket (DO NOT get the tour bus<brown>, take the SHUTTLE bus<green>).
The road from F'banks to Denali can be tough in winter due to blowing snow, and as you get closer to Denali, drifting snow. You could potentially get very very close to the park entrance and then have to stop because of avalanche danger in the narrow passage that the road takes through there.
The hike up Mt. Healy is nice, good views, and accessible from outside the park.
Although Fairbanks doesn't have a large population, it is spread out. There isn't much to the Downtown area, but I suggest you stop in at the APLIC (Alaska Public Lands Information Center) and ask about Denali and local hiking information. APLIC is on Cushman Street (Federal Courthouse). The University Museum has been recently remodeled (and it's not done yet) but is definitely worth a visit. The University is up on a hill overlooking the city. Chena Hot Springs is about 60 miles one way from Fairbanks (there's only one way there and back!) ... there are a number of things to do there, including spending some time in the hot springs themselves. It's quite an experience to sit in that hot water in below zero weather ... your skin is toasty while icicles form in your hair. You could snowshoe/hike up Angel Rocks or, for a longer hike, go to the Granite Tors. There are limited facilities on the Chena Hot Springs Road too (and subject to flooding in periods of high water).
If you're renting a car, then you may not be able to drive to the Whites ... most of the gravel/dirt roads are off-limits per your rental contract. Check to make sure where you can and cannot go (more of an issue in the summer).
Fairbanks experiences a lot of ice fog (smog) during the winter, so be prepared for that. Once you get out of the bowl that Fairbanks sits in, you're okay. Get directions for Chena Dome, and drive up there ... good views of the valley and the city.
Paxson is a good drive from Fairbanks and you go through a lot of "nothing" between "towns".
The Alaska DOT does not use salt on the highways ... they use sand and gravel. So be careful, don't follow too closely or you're going to end up with a cracked windshield (and then you're a real Alaskan!).
Check out Creamer's Field in Fairbanks ... good spot to go xcountry skiing, or skijouring if you've got a dog or two!
There are several good restaurants in Fairbanks ... you might want to check out the Sunday brunch at Pike's (held only in winter for the locals). Or grab a burger at the Chena Pump House.
The folks at Beaver Sports in Fairbanks might also have some information about local hiking/snowshoeing, etc.
It was -44 on Tuesday in Fairbanks, now it's 7 above. Heat wave!!
