Out of FAI, consider an overnight trip to Barrow. Stay at the King Eider Inn, walk the town, take a tour to the beach, and dip your toes in the Arctic ocean. In Fairbanks, there's the University, several good restaurants, and various tourist attractions including a river cruise and a complex which includes a nice air museum. All are worthwhile. But you can exhaust the local possibilities after a few days, so I would think about getting outta town for at least one day. If not Barrow, there are all kinds of hikes etc. to do. BTW, going ANC-FAI, squeeze in Denali if you can. I'm an advocate of the train.