Purchase the Milepost book for $20 in Fairbanks if you want to know what's ahead on the road. Don't count on cell phone coverage!
WARNING: Alaska is remote. Gas stations are irregular, service stations are scant, and several "highways" are actually very rough dirt roads. On some of these dirt roads, semi-trucks drive 60 mph and kick up gravel that can go through your windshield. Several rental car companies will have you sign paperwork saying that you will not drive on certain roads (such as the Dalton Hwy and Denali Hwy).
Denali. The Denali Park entrance is a <2.5-hour drive south of Fairbanks. Vehicles are only allowed 15 miles into Denali National Park; the most spectacular settings are far beyond this and only accessible by buses. All reservations (especially the bus) should be booked in advance. There are "tour buses" and "shuttle buses" -- you want the shuttle bus. (It will feel like a tour anyway.) There is a rigid schedule. I recommend going at least as far as Eielson Visitor Center. This stop is short of Wonder Lake, commemorated in an epic photo by Ansel Adams, but chances are you won't see the mountain like this, though you'll probably see bears, moose, dall sheep, caribou, and more.
Denali Highway. Best road-trip in Alaska (for the effort). Three segments:
Fairbanks-Delta-Paxon (Richardson Hwy, paved, 175 miles, 3-5 hours)
Paxon-Cantwell (Denali Hwy, dirt, 135 miles, 4-6 hours)
Cantwell-Denali-Nenana-Fairbanks (Parks Hwy, 150 miles, 3-4 hours)
The Denali "Highway" is a rough dirt road (see warning message above) that connects the Richardson Hwy in the east to the Parks Hwy in the west. (It is notthe bus road into Denali National Park.)
The trip could be supplemented with a bus ride into Denali National Park.
Don't forget to stop at the Denali fault crossing on the Richardson Hwy, where the pipeline was offset by the 2002 Mw 7.9 Denali fault rupture. There is a pull-out with a low-key exhibit just opposite Milepost 216. Also, further to the south, consider a 1.5-hour hike to the terminus of the Gulkana glacier (ice guaranteed).
Kennecott. 8-9 hours from Fairbanks you will find yourself in Wrangell-St. Elias National Park. 9 of the 13 highest peaks in the U.S. are in these mountains, most of which are not well known. The setting at the abandoned Kennecott copper mineis a breath-taking combination of history and glaciated landscapes. You will pass the world-class salmon fishing locale of Chitina (Copper River) on the way.
Brooks Range. Take the Haul Road (or Dalton "Highway") north to Prudhoe Bay oil fields. This road follows the pipeline and all the truckers hauling equipment back and forth. If you want to see how big and remote Alaska is, this trip is for you. You need several days to enjoy this trip. Plus, things get pretty boring north of the Brooks Range (the "North Slope" is the gradual slope to the northern coast). See warning message above.