The way I would to, is 80 to Wells, NV, go north on 93 through Twin Falls. Then jump on I-84 and at where I-86 goes to the left, take I-86 to Pocatello and jump on I-15. The hwy 26 to 31, through Victor ID and over the Teton Pass into Jackson. However, I'll note that the Teton pass has a 10% grade, but great views at the top and a large area to pull off. And not a problem if your vehicle can pull over a high grade mountain pass and you can downshift coming down.
You can also take I-80 to SLC and jump on I-15 (you only skirt the northern part of SLC so that is great since SLC is terrible to drive through). Once on I-15 cross over into ID and exit onto Hwy 30 to Soda Springs. In Soda Springs go north on Hwy 34 and eventually onto Hwy 89. This will take you over Hoback pass, nothing anywhere near as dramatic as Teton Pass. Get onto Hwy 191 and follow that north to Jackson.
Really depends on if you have no problem with steep grades.
Trailer: Lifted 228BH, heavy duty springs and Yokohama tires
TV: 2016 GMC Sierra Z71 4x4 CC, SLT
Spare TV: Two Alaskan Malamutes
Living somewhere in ID; previously lived in Moab UT; previous to that, don't ask!