Welcome aboard the RoadTrip America forum.
Mark is correct in noting that the "best" route depends on the time of year. It also depends on your own tastes, what you consider relaxing, and what type of scenery appeals to you. I'm going to assume that since you posted in the Planning Summer Road Trips forum, that you'll be travelling this summer. And then I'm going to tell you what I would do if I had a week to make the trip. Basically, I'd head across southern Utah and Colorado hitting one gorgeous national park after another.
Such a trip would have you start out by heading up to Las Vegas for a day or two, not so much for the shows and gambling as the great parks nearby such as Red Rock Canyon and Valley of Fire. From there, continue up I-15 into southern Utah where you'd use UT-9 into the Canyon area of Zion National Park. Continue eastbound on 9 to US-89 north for a short jog over to UT-12 east and Bryce Canyon National Park, and then beyond to UT-24 and Capitol Reef National Park. At Hanksville, take UT-95 south to Natural Bridges National Monument. From here take a short detour down UT-261 and US-163 through Monument Valley, then pick up US-160 east through Four Corners to Mesa Verde National Park. If you're of a mind, take a nostalgic ride on an old narrow gauge steam railroad.Continuing east, you'd go up and over the 10450 foot high Wolf Creek pass. Eventually, you join up with I-25 for a short run down to Raton, NM where you pick up US-64 to Capulin Volcano National Monument and a hike around the rim (you can drive to the top of the caldera). Continue on US-64 and US-87 into Amarillo. From there US-287 basically follows the Red River Valley into Dallas. With all those parks to visit, be sure to pick up a National Parks Pass. The most amazing part about this trip is that, at only 1800 miles, it's just a few hundred miles longer than the shortest, direct interstate route between LA and Dallas of about 1450 miles.
AZBuck