The drive from Denver to Yellowstone National Park and Grand Teton National Park takes at least nine hours and that does not take in possible road construction. You may want to revise your itinerary and forgo the Denver area and flight and fly directly into Bozeman, MT. From this point you are with in easy driving distance to visit the two parks.
I would suggest you use Gardiner MT the north entrance to Yellowstone as your first base, it is 1½ hours from the Bozeman airport through Paradise Valley. In Gardiner, you have the opportunity to horseback ride, raft the Yellowstone River, and be in close proximity to some of the best wildlife viewing in Yellowstone.
From Gardiner it is just a short distance to the Tower/Junction, Lamar Valley area.
Here you have the chance of seeing gray wolves, grizzly and black bear, bull elk with antlers in velvet, big horn sheep, mountain goats, moose, pronghorn, and many other species of wildlife. You may want to consider the services of a local guide for you first day, as they know the area, the best place to locate the wildlife, and carry binoculars and spotting scopes to help with viewing.
There are many campgrounds near Gardiner, either in the Gallatin National Forest or in the Yellowstone National Park close to the north gate entrance. There are also many hiking trails in the north end of Yellowstone ranging from easy class to strenuous class.
After 2-3 days if you chose to relocate I would recommend basing out of West Yellowstone, MT. It is the west gate and is the closest access to the lower end of the loop system and the closest to Old Faithful and the many thermal basins.
From West Yellowstone, you may then want to head to Jackson, Wyoming and the Grand Teton National Park. The Jackson Hole chamber can help you with any booking information you would need. From Jackson you could drive into Idaho and return along the west side of the Teton Range, about 5 hours