Another suggestion to consider, when you make it over to this side of the state, take the train to Seward, pend the night and return back to Anchorage. We've never done this before but it is high on our list of things to do in Alaska.
We have done the train to Denali fron Anchorage and stayed at one of the lodges and it was a relaxing trip. You could probably do the train from Fairbanks to Denali then on to Anchorage and pick up a rental car.
