OK. Here is a suggestion.
Day 1: Head directly to the Old Faithful Visitor Center, skipping West Thumb which you will pick up on your way out. Check on eruption times for the geysers that are predictable. Write the times down on the brochure available at the VC desk for 50 cents. Walk thru the geyser basin.
Day 2: If you wake up early, take another walk thru the geyser basin before breakfast. Then head north toward Madison Junction stopping at Midway, Firehole Lake Dr. , Fountain Paint Pots (when you exit Firehole Lake Dr you are right across from the entrance to Fountain Paint Pots. You can't go straight across, but you can turn left or right and turn around in the nearest pull out). Do Firehole Canyon Dr, which will take you back toward the South, but just for a couple of miles. It is beautiful in there. Stop at Madison Jct for a pit stop...flush toilets! Head toward Norris, then on to Canyon Visitor Center. Do the South Rim viewpoints and the North Rim viewpoints of your choosing. At least do Artist Point on the south rim...it is the easiest to access and the best view point. Head south again and go thru Hayden Valley stopping if you see anyone with cameras with long lenses or spotting scopes. Do Mud Volcano and Le Hardy Rapids if you have time. Return to Canyon.
Day 3: ***Not sure exactly what day you are getting to Yellowstone, so be aware the road from Tower Jct to Dunraven Pass will close on 9/22 for road construction. If you are there the 22 or after, you will need to get to Mammoth via Norris.*** Head North over Dunraven stopping at Tower Fall and some of the overlooks north of there unless they are closed for the road construction. At Tower Roosevelt Jct, turn right in to Lamar Valley. Again look for wildlife or long camera lenses. I usually go to Soda Butte and turn around and head back to Tower Roosevelt Jct. (If you do stop at Soda Butte, be sure to walk around the back side and look at all the swallows nests!) At Roosevelt, head toward Mammoth. Take the drive back to Petrified Tree, then do Blacktail Plateau. Many see wildlife back in here, alas I never do! (Blacktail Plateau is dirt/gravel, so if you have a rental car you may not want to take it. I have 4WD, but regular passenger cars can safely navigate this.) When you get to Mammoth walk the lower terraces. The good bathrooms here are the new ones by the bus parking lot and the gas station. Go to the Visitor Center (really yucky bathrooms on the lower level accessed by stairs out front). Do the walk of Old Fort Yellowstone if you wish.
Day 4: Finish anything you missed in Mammoth. If you want, head north to Gardiner to the Roosevelt Arch. Look for bighorn sheep on the cliff on the right. Head south from Mammoth toward Norris. Do the Upper Terrace Drive on the right as you climb out of Mammoth. Do the Norris Geyser Basin. Head to Canyon, doing Virginia Cascades Dr if it is open. Stop at Canyon VC for a pit stop (flush toilets). Head south thru Hayden. At Fishing Bridge, stop and walk across. You are not looking at the bridge, you are looking for wildlife!
Day 5: Go south from Lake around the lake. Enjoy West Thumb geyser basin on the way out. There are some short hikes in the Lake area if you want to get in a walk before you leave the park. Natural Bridge is easy (on an old road bed) and Storm Point which is east of Fishing Bridge.
BTW, just because you have an itinerary doesn't mean you have to stick to it if something more interesting comes up (mama bear and cubs, wolves, pikas (lol), awesome geyser eruption, etc). It's just a guide to head you in a direction from your lodging with an indication of what you might see for the day.