Head up to Jackson via Evanston then following the ID/WY/UT border. Head out of the park via West Yellowstone and head to Sun Valley area. Visit Arco, ID (first nuclear reactor museum), Craters of the Moon then Sun Valley. From there you can do sidetrips in the area including up to Stanley on the Salmon River.
You can exit down to Twin Falls and follow I-84/15 back to Salt Lake. Antelope Island near Salt Lake might also be a nice finishing excursion.
I hope I'm not chiming in too late on this thread, but Tumbleweed has suggested a good itinerary. I definitely second the suggestion of Stanley and the Sawtooths.
As a pug owner for 13 yrs. I appreciate your screen name! Thanks for the suggestion!