Beartooth & Chief Joseph Scenic Byway
You've heard of the Beartooth Hwy that begins south of Red Lodge, Montana, but there's another road that's not to be missed - Chief Joseph Scenic Byway, Hwy 296 north of Cody, Wyoming (see map). Here's another link to more Red Lodge info, and Beartooth photos from Tony Taylor's 2012 Wild West Tour.
The Beartooth Hwy near Red Lodge, Montana is probably the best switchback road in the US. See map with directions.
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Swan Valley, ID is 260 miles from Salt Lake City, making this about 1120 miles total. it includes the Beartooth Hwy and
Chief Joseph Scenic Hwy (see above) plus Yellowstone NP, and both east and west views of the
Grand Tetons. With it's 40 degree constant temps, Minnetonka Cave might be a nice place to stop and cool off.
Jackson, WY is famous for the elk antler arches on the town square. Two restaurants on Broadway have been featured on Food Network's Diners, Drive-ins, and
Dives. Or try the stuffed pretzel at Sidewinders (also seen on DDD), a few blocks west.
"Jackson wy" by Monster1000 - Own work. Licensed under CC BY-SA 3.0 via Wikimedia Commons - https://commons.wikimedia.org/wiki/File:Jackson_wy.jpg#/media/File:Jackson_wy.jpg
This 1:23 video trailer is a portion of a video available for purchase from http://www.yellowstonenationalpark.com/wondersDVD.htm
Old Faithful geyser is in the SW corner of the loop. Most geysers are north of Old Faithful. Upper and Lower Falls on the Yellowstone River are just south of Canyon Village, in the NE corner of
the park. Here's the website, where you can zoom in closer on the map. Click on "View Park Map" in the left sidebar. Note the close-up maps at the
top.
See the website for an interactive version of this map. The map is down the page.