Three types of buses travel the Denali Park Road. At the high end are the tan buses, guided tours led by experts who provide running commentary throughout the drive. Passengers stay with the same bus and driver throughout their tour, getting off only at rest stops. Least expensive are the camper buses, which are for the benefit of tent campers traveling to and from their reserved camp sites in the interior of the park. In the middle are the green shuttle buses: purchase a round trip to one of several possible turnaround points: Toklat River, Eielson, Wonder Lake, or Kantishna. You can get off and on the shuttles as often as you'd like to hike, explore, or even take a multi-day trek. When you're done, you simply go back to the road and flag down any green bus passing by in either direction, and continue your journey where you left off.
If you go:
The park buses are operated by a private contractor, Aramark, and all of them begin their run at the Wilderness Access Center, near the main entrance. On any given day, the earliest departures are best, if you want to improve your odds of joining the "30% club". (That's the proportion of overall visitors who actually get to see the mountain, when it's not hidden by clouds).
Links:

Bus Stop
Park Road
Denali National Park and Preserve, Alaska 99755 USA
Mailing address:
Denali National Park
P.O. Box 9
Denali Park, AK 99755 USA
Phone Numbers:
Aramark Reservations
+1 (866) 761-6629
+1 (866) 761-6629
Denali Visitor Information
+1 (907) 683-9532
+1 (907) 683-9532
Message from RoadTrip America