Bryce Canyon National Park spreads across Garfield and Kane Counties in South-western Utah. It lies on UT-63 about 24 miles southeast of Panquitch. It is a spectacularly scenic route and driving down has its moments. To hit UT-63 from Salt Lake City in the north, you would need to go south on I-15 and get off at exit 95 into UT-20. Continue east till you hit UT-89 and then turn south passing through Panquitch to UT-12. Keep going till you hit UT-63 and you are there.
From the South, I-15 takes you north to UT-9 and Zion National Park. Once you're done with Zion keep going till you hit UT-89 at Mt. Carmel Junction, from where you head north to UT-12. Then you turn south on hitting UT-63 to turn into the Park entrance. If you're driving in from Grand Canyon National Park which is about 290 miles away, you'd turn into AZ-64, make a left onto US- 89 for about 216 miles after which you'd need to turn right into UT-12. When you're just under 14 miles on, you hit UT-63 and you're there.
The Park encourages you to leave your vehicle outside and use the free shuttle service to avoid congestion and ensuing ill effects. This actually makes sense because you do not have to worry about finding parking space or making your own way around without getting lost or losing kids. Of course, if you're bent on driving in, you may do so, after paying a fee.
If you're flying in, you'll be landing at Las Vegas, Nevada, or Salt Lake City. From these points it is about 270 miles to the Park. You could even wind up in Cedar City or St.George which are smaller airports and comparatively closer at 80 miles and 125 miles respectively.