I think it depends on time, if you only have a day for Yellowstone, then hit Norris Geyser Basin and the Old Faithful areas, you will miss the falls and the Mammoth area, but you will see much of the options available in the park so far as geothermal is concerned, if you have a full day, or more, hit the outer loop, making the stops.  Other than Yellowstone, there is Jackson just south of there and will take you back to I-80 or Salt Lake City, also on 80.  From there it is easy to plan your trip.  Of course, if you go East out of Yellowstone, there is Devils Tower National Monument and all the old west towns in Wyoming, then not much further to Mt Rushmore and Crazy Horse Memorial, lots of options within your route.

Many options you have to decide if you want an "American Safari" looking for Grizzly Bears, Wolves, Bison, Elk etc. or if you want the physical wonders of Yellowstone.

If you want a safari go through the Lamar Valley bring your binoculars at dawn and dusk.  The two best ways to enter the park for this area are the NE entrance through Cooke City, MT or the east entrance through Cody, WY.  Cody is a neat spot with a bit of western history of its own.  Hayden Valley is also a great spot.

From Jackson Hole you would go to the South entrance through Grand Teton National Park and John D Rockefeller Jr Pkwy incredible sights on its own.  You should go to Old Faithful Geyser and check out the geysers and take a dip in the Firehole.  Go to the center of the park and see the falls and exit through Madison for the thermal features.

If you really want to experience Yellowstone 99.5% of people spend their time within a mile of the roads.  Get a backcountry permit and go for a backpack.

Howling Coyote in Livingston has great green chili if its still there.

Have a blast.