My son and daughter-in-law were park rangers at Yellowstone so I've been there a number of times. You've got some good information above, so I won't add to it. The web site will give you most of what you need to plan your trip. Bozeman is the closest 'major' airport, and it's an easy drive down. Do the touristy stuff, of course, but that should only take a couple of days. My daughter-in-law (whose name is, ironically, Dove) runs private tours of the back country through Yellowstone and Grand Teton. That kind of tour will get you to parts of the park that are not usually seen and see animals that you don't usually see as well.
Spend a day in Jackson. Great town! Be sure you have a beer at the bar where all the bar stools are saddles. Check out the old west gunfights. Raft down the Snake River. Also, (it's a little bit of a trek - maybe 3 hours) stop off at the Little Bighorn Battlefield.
Just don't treat the animals, especially the bison, bear and elk, as tamed zoo animals. Lots of people get hurt being stupid. I once saw some silly SOB taking his 4 year old son by the hand and venturing out very near a herd of bison. The bison, of course, were rutting and had calves nearby. The mommas were not amused.