How much time do you have ;-)
I ask that semi-facetiously, but my "not to miss" itinerary would be different for different lengths of stay in the Park. So I'll prioritize.
Priority 1: Upper Geyser Basin ("Old Faithful" area -- find out what's going to erupt of the predictable geysers. If at all possible, see an eruption of Grand Geyser. Great Fountain predictions are at the Visitors Center by Old Faithful, but it's in the Lower Geyser Basin - a 15-30 minute drive away. Great Fountain eruptions can last an hour and they are better in the middle than at the beginning!)
Grand Canyon of the Yellowstone
Mammoth Hot Springs
Priority 2:
Midway Geyser Basin
Lower Geyser Basin
Norris Geyser Basin
Tower Falls
West Thumb/Yellowstone Lake Area
Hayden Valley
Priority 3:
Firehole River Canyon (one-way loop drive on the way from Old Faithful to Madison)
Mud Volcano area (near Hayden Valley)
Pocket Basin/Ojo Caliente (natural area, BE CAREFUL!)
Obsidian Cliff and Golden Gate Area (on the way from Madison to Mammoth Hot Springs)
Le Hardy Rapids (just a few miles north of Fishing Bridge, somewhat at the mouth of Hayden Valley)
That's a good start. Some of the best backcountry views are on the drive from Tower to the Northeast Entrance.
Also, check out this link:
If we plan to do that on day one, do you think we would have time to see much else that day and then go back another day to see the rest?
I guess I need to look at the maps on the website you provided the link to and try to plot out an itinerary to squeeze as much as possible.