Here is a suggestion for an itinerary. You may want to flex it some based on what you want to see and what you do see! Be sure to ask at the entrance gate if any roads are closed. Some have been closed and opened this week with smoke issues. You may have to adjust how you get to things based on that.
Day 1: Enter the NE gate. Stop in Lamar if you see people out with cameras or scopes. At Tower/Roosevelt Jct, turn left and head toward Tower Fall. Stop and take a look. Have ice cream, lol. Continue south over Dunraven Pass, turn right at Canyon and head for Norris, Madison and then to Old Faithful. Overnight at Old Faithful
Day 2: Make this your geyser day. Start at the Visitor Center, walk the Upper Geyser Basin. Write down the eruption times for the predictable geysers on the back of the brochure at the VC (50 cents) and try to work some of them in to your walk. I like to eat lunch at the Lower General Store which is to the right side of the Old Faithful Inn if you are standing with your back to the geyser basin. There are gas pumps out front. After lunch, finish looking around there, then jump in the car and head toward Madison Jct stopping at Midway Geyser Basin, then turn in to Firehole Lake Dr on the right. At the exit, go right toward Madison again, then take Firehole Canyon Dr. If either of you need a pit stop, go on to Madison Jct as they have nice flush toilets there. Head back toward Old Faithful, taking Firehole Canyon Dr. Stop at Fountain Paint Pots and Biscuit and Black Sand Basins if you wish. Get ice cream on the 1st floor of the Old Faithful Inn and take it up to the 2nd floor deck overlooking the geyser basin. Take a look thru the Old Faithful Inn afterward.
Day 3: Head down to Grand Tetons for the day.
Day 4: Make this your Canyon day. Head to the Canyon Visitor Center and tour that, then do the North and South viewpoints as you desire. The easiest viewpoint on the South rim is Artist Point. I also like Brink of the Upper Falls on the North rim if you just want to do one each. There are brochures at both the Visitor Center and at the trail heads to help you pick. Head south thru Hayden Valley, stop at Mud Volcano, LeHardy Rapids and Fishing Bridge. Be sure you park and walk across. You are not looking at the bridge, but you are looking for wildlife in the river and along the sides of the river...or in the trees! Have ice cream at the Fishing Bridge General Store, lol. Return to Old Faithful either by continuing south around the lake or going back thru Hayden Valley and Canyon.
Day 5: Do you need to catch your plane today? If so, I would just head straight to Norris and out thru Mammoth without stopping. If you don't have a deadline to get to Billings, you can stop at Mammoth and look around the terraces. Drive the Upper Terraces which will be on your left as you start downhill from Swan Lake Flat, and walk some of the boardwalks for the lower Terrace. If you have time, get the brochure from the Visitor Center and take the walk around Old Ft. Yellowstone then....have ice cream at the Mammoth General Store!
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.
Have fun and we would love to have a trip report from you when you get back!