Drive north on I-15 from Salt Lake City, to Idaho Falls 3 hour drive. From Idaho Falls east on US 26 to Swan Valley, 30 minutes.
In Swan Valley you can continue to Jackson via US 26, or turn left on Idaho 31 up to Victor, turn right onto Idaho 33,(turn left to go to Targhee) which will turn into Wyoming 22( Teton Pass)another 45-60 minutes at most, provided the pass isn't closed. You can stop in Swan Valley, and ask any local if Teton Pass is open
Be sure to drink a few pints of Teton Ale, a great microbrew, brewed in Jackson
And if you have a extra day in the Salt Lake area, drop me a line, i'll help you out with some passes, to a resort here.