MO,
As you can see by the variety of answers, Yellowstone is HUGE. At nearly 3500 square miles, its larger than 11 states and occupies part of 3 states. One lap around the Grand Loop road is 140 SLOW miles. Lots to see. And the part of the loop that goes by Old Faithful and the geyser basin is closed until June 11th due to a road collapse. YNP sits on top of a huge underground lava field.
Pack clothing for every season. YNP park is 8,000 ft. ASL. Missouri averages 500 ft. ASL. You'll get acclimated to the altitude since you're driving, but don't over exert yourself.
Take lots of pictures and spend the drive home planning your NEXT visit. And on that trip, spend some time in Grand Teton NP. Even though they're next to each other, the Tetons are completely different. I actually prefer it to Yellowstone. The scenery is so spectacular, my blood pressure drops about 50 points! I've been known to just sit and stare. :)