You need to make the trip to the salt a trip you will always remember.
Take 15 north out of LA to I40. I40 east to Williams, AZ. Then 64 north to the Grand Canyon. Then continue east on 64 east to 89. North on 89 and south of Page, AZ take Alternate 89 past Marble
Canyon/Vermillion Cliffs to Fredonia, AZ and then north to Kanab, Utah where you will reconnect with 89.
North on 89 at Mount Carmel Junction go east 26 miles on 9 into Zion National Park. Then back track on 9 back over to 89. Continue north on 89 to 7 miles south of Panguitch, UT and then east 14 miles on 12 to Bryce Canyon National Park.
Then either back over to 89 or stay on 12 and head north through Escalante and up to Torrey where you will hit 24. There only 16 miles east is Capitol Reef National Park. But mainly at Torrey head west on 24 till you get to I70. Don't get on 70, but head north on 28 for about 7 miles till you get to 50. Take 50 northwest until you get to I15. Head north towards Salt Lake City.
If you want to by-pass Salt Lake City on a pretty route (I go this way) head west at Santaquin, UT on 6 over past Eureka where you will get 36 north. About 44 miles north there is a road that goes west a couple miles over to Rush Valley. Just as you come into what is Rush Valley (not much) take what looks like a local road north. It will take you north to Grantsville. Turn left onto 138 there and it will take you out to I80 and it is west there to Wendover. A long stretch without gas so make sure you have plenty before leaving Grantsville. I fill up at Eureka and have plenty to get to Wendover.
If you take this route you will see some of the most spectacular south west scenery you can see. You have mountains and trees at home. You need to see "canyon country". You won't regret it.
c ya,
Sum