From SLC go east on I-80 until just past Evanston, Wy about 8 miles E until US 189. US 189 will take you to Jackson. You could also go I-15 to Idaho Falls, then east on US 26 to Hoback Junction, then north to Jackson. It's a longer route than the first one.
DO NOT try Idaho Falls ID 33 E that turns into Wy 22 to Wilson, Wy. It goes over Teton Pass, 10% grades on both sides of the pass.