What to do with one day at Yellowstone?

If you want to do the North American Safari head to Lamar Valley east of Tower Junction a good hike to Tower Fall exists near by.  Another spot for wildlife is Hayden Valley, but you may encounter any type of wildlife at any spot.  
There are epic spot throughout the park natural man made it's incredible.  Check out Yellowstone Falls out of Canyon Village.  Link for a few grand waterfalls in the park http://www.yellowstonepark.com/2014/01/3-waterfalls-of-the-grand-canyon-of yellowstone/ and
http://www.nps.gov/yell/planyourvisit/ncanyon.htm (actually NPS websites are great for planning, unlike most park or recreation sites).
Of course the legendary Old Faithful Inn and Roosevelt Lodgeare pretty incredible buildings and they survived the fires back in the late eighties.
Check out the grand Old Faithful is a must.  If you wanna cool off go to Firehole River and Firehole Falls.  Fly fishing is great where ever you go.  
Exiting the park I'd go South Entrance so you can see Grand Teton National Park and travel the John D Rockefeller Jr Pkwy.  Grab food in Jackson (Wyoming).  Or Northeast Entrance Station which goes through Cooke City (Montana) and you can take the highway that goes outside the park to Cody.  Lastly you can go out the East Entrance - Yellowstone National Park to Cody.   I don't like West Yellowstone and Mammoth Hot Springs entrances.
Have fun and honestly buzzing through Yellowstone is an awful way to do it.  You will encounter crowds and traffic.  Plan on early morning and evenings.