According to mapquest, this is the shortest route, and you don't go through vegas! The directions look funny here, click the link in the source
1: Start out going WEST on E 400 S / UT-186 W toward S MAIN ST. 0.1 miles Map
2: Turn LEFT onto S WEST TEMPLE / UT-270. 0.1 miles Map
3: Turn RIGHT onto W 500 S / UT-269 W. Continue to follow UT-269 W. 0.8 miles Map
4: Merge onto I-15 S toward LAS VEGAS / CHEYENNE. 84.4 miles Map
5: Take EXIT 222 toward I-70 / SALINA / RICHFIELD. 0.5 miles Map
6: Turn SLIGHT LEFT onto I-15 BR / UT-41 / UT-91 / UT-28. Continue to follow UT-28. 8.8 miles Map
7: Turn LEFT onto N MAIN ST / UT-28. Continue to follow UT-28. 38.1 miles Map
8: UT-28 becomes US-89. 8.0 miles Map
9: Merge onto I-70 W / US-89 S / UT-70 W toward RICHFIELD. 32.9 miles Map
10: Merge onto US-89 S via EXIT 23 toward PANGUITCH / KANAB. 61.7 miles Map
11: Turn LEFT onto E CENTER ST / US-89. Continue to follow US-89. 67.5 miles Map
12: Turn LEFT onto E 300 S / 300 ST S / US-89. Continue to follow US-89 S (Crossing into ARIZONA). 202.2 miles Map
13: Turn SLIGHT RIGHT onto ramp. 0.1 miles Map
14: Turn SLIGHT RIGHT onto I-40 BL. 0.4 miles Map
15: Merge onto I-40 W / PURPLE HEART TRL. 5.0 miles Map
16: Take EXIT 195 toward AZ-89A N / FLAGSTAFF / GRAND CANYON. 0.1 miles Map
17: Merge onto I-17 S via the exit on the LEFT toward SEDONA / PHOENIX. 138.7 miles Map
18: Merge onto I-10 E via EXIT 200A toward TUCSON. 2.6 miles Map
19: Take the 7TH ST exit- EXIT 145A. 0.2 miles Map
20: Keep RIGHT at the fork to go on N 7TH ST. 0.9 miles Map
21: Turn RIGHT onto E WASHINGTON ST.