One of the highlights from my trip to the US was Yellowstone and Grand Teton! Yellowstone I think is so popular because it literally has everything in it - old volcanic springs, geothermal areas, a giant canyon and waterfall/river, beautiful meadows of wildflowers, forests and plains filled with wild animals. It's literally got everything and everywhere you go there is something exciting to be seen.
We camped at Yellowstone two nights in the summertime - its as an EXTREMELY busy time and we were lucky enough to get a campsite because one freed up while we were at the airport in Sydney so I booked it straight away - normally sites book out 5 months before the peak period, so be careful arriving without a campsite! I'm really glad we found a spot inside the park, saving us a lot of time getting in and out of the park everyday. The campsites were not huge but decent with good facilities; it cost a bit though!
The animal sightings were definitely a highlight of Yellowstone- you don't need to worry about this at all: you will see animals. They are on the side of the road, in the the meadows, even in the sulfur pits! It's impossible to miss them! If you drive out to Lamar valley you will even see massive herds of bisons and pronghorns or elk. Really really amazing.
If you can only make it to one national park, it's gotta be Yellowstone. I know it's not the easiest to get to (Wyoming), but it's truly worth it and breathtaking. Let me know if you have been, or plan on going!