Fairbanks is the second biggest city in Alaska. It's spread over a wide area so tiring to walk around, but if you have time there's plenty footpaths and sidewalks. The Morris Thompson visitor centre down town can give you ideas for things to do and ways to get around, though there's not a huge amount that's cheap or free.
Hitching in and near Fairbanks can be a great experience - in general the further from Fairbanks you got, the worse hitching! You can usually catch a lift in the Fairbanks area in less than 30 minutes. Some even got picked up as late as midnight one night (a bright Alaskan summer midnight but even so).
West towards Esther you can stand at one of the roundabouts near the University, walk as far out of town as you like really. To head north there are intersections and sliproads on the road to Fox, and east towards the Alaska Highway it's a fairly easy walk from the centre of town to the start of the Richardson Highway.
If you get out onto highways it is worth standing by the side of the road, as long as you're safe, as highway drivers will stop and cops don't seem bothered. trash:Fairbanks