At first thought, West Yellowstone pops to mind, but with another look, it might be easier with you as the sole driver to stay in Mammoth or Gardiner. It will be less driving for you plus it would position you for an early AM to Lamar and you might be able to catch a Ranger program at Mammoth.
Monday: Driving from Missoula, you can probably be to West Yellowstone by lunch or shortly thereafter. I drive from N. Idaho and I always exit the Interstate at Cardwell. After you make the right turn at the bottom of the exit ramp, the big brown sign routes you to the left. This goes to Lewis and Clark Caverns if you want to see that. I usually go straight onto the smaller county road which cuts a few minutes off the journey. This connects with US 287 on down the way. This will take you in to West Yellowstone. Have your son start looking out for pronghorn antelope along the way!
Once you get to West Yellowstone, he might enjoy the Grizzly and Wolf Discovery center that afternoon which is next to the IMAX and across from McD's. You can also stop in the West Yellowstone Visitor Center for Yellowstone and get your entrance paid, then you can go thru the short line at the gate the next morning. This will also give you a chance to look at the little newspaper handout and see if there are Ranger programs of interest to you. They are divided by location. If you are anxious to get in to the park, drive in to Madison junction and turn right. Cross the Gibbon River and turn right in to Firehole Canyon. Rejoin the main road and continue toward Old Faithful. Stop at Fountain Paint Pots, then a bit further on the main road and do Firehole Lake Dr. This will probably be enough windshield time! Return to West Yellowstone before dark.
So, here's how I would send you if you are spending night 2 in Gardiner.
Tuesday: Head in to the park. After about 5 miles have your son keep his eyes peeled for bald eagle in the trees. You might also see a trumpeter swan on the river at about the 7 mile mark. After you cross the little bridge, you will likely see elk and bison. At Madison, turn left and head for Norris Jct. Stop at Artist Paint Pots if the parking lot is not full. Do Norris Geyser Basin, then head north to Mammoth. Do the Upper Terraces by car, then walk the lower terrace. The best bathroom in this area is right by the gas station at the edge of the bus parking lot. The Visitor Center is a bit further on, and is interesting as well. They have a self guided tour of Ft Yellowstone and maybe you can check on the ranger programs for the afternoon/evening.
Wed: Head East to Tower Roosevelt Junction. Turn left toward Lamar Valley. Look for wildlife, stopping if you see people stopped with big camera lenses. Return to Tower Jct, turn left and go to Tower Fall. Stop at some of the overlooks if you can...there is roadwork going on along here, so not sure what the situation will be. After Tower Fall, cross Dunraven Pass, stop at the Canyon Visitor Center. Do overlooks on the North and South Rims. If you are wanting to cut this short, I would do the Brink of the Upper Falls on the North rim and the Artist Point overlook on the South Rim. From here head south into Hayden Valley again hunting for wildlife. Stop at Mud Volcano. Continue South, stopping at Fishing Bridge. Park and walk across and look for wildlife here as well. Overnight at Lake.
Thursday: This will be your geyser day. From Lake head South to West Thumb and then to Old Faithful. I would skip West Thumb geyser Basin and do it in the AM as you head to Grand Tetons as you will come back to this junction to head south. At Old Faithful, head to the VC. Write down the predicted eruption times on the brochure available at the desk. (50cents). If you are early enough, take the Ranger walk (I think it's at 830). Otherwise, head around and behind Old Faithful, I usually go to the left, then cross the bridge directly behind OF and go to Geyser Hill. Tour the geyser basin down to Morning Glory Pool (pit toilet here) then return to the main area by the paved walk. I like to eat lunch at the Lower General store which is by the gas pumps and fronts on the geyser basin. Save room to eat ice cream on the 2nd floor deck of the OF Inn overlooking the geyser basin. One of you must get huckleberry. It is a rule. LOL!! Go back to the main road, head north to the Fairy Falls trailhead, hoping there is parking room! Get your picture of Grand Prismatic. Back in the car, head north to Firehole Lake Dr and Fountain Paint Pots if you didn't do these on Monday when you got in to West Yellowstone.
Friday: Leave Old Faithful, stop at Keppler Cascades if you wish, then West Thumb geyser Basin if you are not geysered out!