OK, here is one suggestion:
Day 1: Arrive IF, drive to West Yellowstone and enter the West Gate. Head to Canyon. It will take you about 1.5 hours to get to Canyon from West without stopping. How much you see will obviously depend on what time you get in to Idaho Falls. It's about 2 hours from IF to West. Things to see would be Artist Paint Pots (short trail), Canyon Visitor Center, take a short drive down into Hayden Valley looking for wildlife, some of the overlooks for Canyon. The most accessible one is Artist Point. Overnight at Canyon
Day 2: See the rest of Canyon if you didn't see it yesterday. Head north from Canyon over Dunraven Pass. Do Tower Fall and some of the overlooks to the North. At Tower/Roosevelt Jct turn Right and head in to Lamar Valley to look for wildlife. I would turn around at Soda Butte. There is a sign here and you can tell when you get there. Go back to Tower Jct, turn right and go to Mammoth. Do the lower Terrace, then drive the Upper Terrace. Head south toward Norris. If you have any time, do a short stop here. You need to leave here at least an hour before sundown to get to West Yellowstone before dark. This will be a very long day. Do as much of Canyon as you can the afternoon before. Overnight in West Yellowstone.
Day 3: Make this your geyser day. Head to Old Faithful. I usually recommend people skip the Lower Geyser Basin and Midway GB and head directly to Old Faithful, but with your time constraints, I recommend you do things as you come to them. After you make the right turn at Madison Jct, cross the Gibbon River Bridge, then do Firehole Canyon Dr, Fountain Paint Pots, Firehole Lake Dr. (on the left afer Fountain Paint Pots), then Midway. After that, go to Upper Geyser Basin and do the boardwalks there. It is about 36 miles or so from Old Faithful area to Lake. In Yellowstone you double your mileage to get predicted time, so count on about 1.5 hours from OF to Lake. Overnight at Lake
Day 4: This morning, drive north to Fishing Bridge. Park and walk across. Go north again, do Le Hardy Rapids and Mud Volcano. If you want, drive into Hayden and take another look-see, then head back south toward Grand Tetons.
You will do better if you stop in Idaho Falls and pick up a cooler and some picnic fixings. You can keep on the move and not have to find a place to eat lunch. I do recommend ice cream at the Old Faithful Inn. Get it on the 1st floor, then take it up to the 2nd floor deck overlooking the geyser basin.
Here is a link to sunset in the park. It is not recommended to drive after dark in the park as the big black animal shapes take over the road!
This is a quick overview of the park. Many of these days will be long, but if you are traveling by yourself you can probably do it! Not figured in to the mix are the potential for bison or bear jams.