OK, so here is what I would do with the days you have:
Day 1: Travel from GTNP to Old Faithful. Tour West Thumb on the way in and the do the Upper Geyser Basin around Old Faithful that afternoon. All the area is boardwalk or paved, so easy walking. Be sure to enjoy either an evening drink or an early morning coffee on the 2nd floor deck of the OFI overlooking the geyser basin. As THBergs indicates, start at the Visitor Center so you can check time for the predictable geysers. I think there is usually a geyser walk about 5 or 530. There also may be a ranger program that evening and they are always interesting.
Day 2: Finish up the Upper Geyser Basin, then as you leave the Old Faithful area, head north toward Madison. Work your way thru Midway, Firehole Lake Dr, Fountain Paint Pots and Firehole Canyon Drive to Madison Jct. (Nice rest area here. Flush toilets! lol) You can drive partway out the west entrance road if you want. There are usually lots of elk out this way in Sept. and it is a pretty drive along the Madison River. Turn around when you cross the bridge so that the river is now on your right (north side of the road). Head toward Canyon, stop at Gibbon Falls, do Artist Paint Pots if you have time. Take the Virginia Cascade Drive. Stop at Canyon Visitor Center. Depending on how much up and down walking you want to do, the easiest viewpoint for the falls is Artist Point on the South Rim. The walk is pretty level and it is my favorite view. Continue to head toward Lake, driving thru Hayden Valley, doing Mud Volcano, Le Hardy Rapids and Fishing Bridge. Park and walk across Fishing Bridge and look for wildlife. You might see otters!
Day 3: Head back toward Canyon. In Yellowstone it does not matter if you backtrack. You just increase your chances to see wonderful wildlife! At Canyon, turn left. Stop at Norris Geyser Basin, then work your way north toward Mammoth. Stop at Sheepeater Cliffs if you want to see some nice columnar basalt. Drive thru the Upper Terraces. Go to the Visitor Center, walk around Ft Yellowstone if you wish, then go do the Lower Terrace. The best bathrooms here are the new ones by the bus parking lot between the Liberty Cap formation and gas station. The ones at the Visitor Center are old, down stairs in the front and very small. If you have a lot of daylight left, you can head out to Gardiner and drive thru the Roosevelt Arch. Look for bighorn sheep on your way out. If you have the book Yellowstone Treasures, she tells you exactly where to look for them. On your way back into the park, you will cross the bridge so river is now on your right, and there are a couple of pull offs. Pull to the front of the paved one and then get out with your binocs and scan the cliff area. Depending on how your time is running, either return to Canyon by way of Norris or go to Tower junction and down.
Day 4: Finish up anything at Canyon you missed, then head north over Dunraven Pass. Stop at Tower Fall, then stop at the overlooks north of the Falls. Be sure to look behind you and across the road up on the cliff area to see if you can see bighorn sheep. At Tower Junction, head out in to the Lamar Valley to look for wildlife. Return to Tower. Turn Right as if you were headed to Mammoth, do Blacktail Plateau Dr. This is dirt and gravel, but usually OK if you take it slow. You may spot some good stuff back in here. Blacktail Plateau comes back to the main road right by Petrified Tree, so duck in there as well. Return to Canyon.
Day 5: Check the weather at the Canyon Visitor Center for Beartooth Pass. If the Ranger tells you it's good, head North over Dunraven again, then out thru Lamar and on out the NE entrance to Cooke City and Beartooth. If the weather is not favorable, you will head out the North entrance via Gardiner to Billings. The important thing here is to have your back up plan as PeachBelle suggests. I am guessing you will have paper maps which is imperative!
Here is a link to the park map.
Did we nag you to get Janet Chapple's Yellowstone Treasures? It is wonderful for planning and for bringing with you. She has a wonderful mile by mile road log.