I've made the drive from south Texas to LA and back a few times. It's a boring drive. 12 hours just to get out of Texas. Same rolling view the whole way. You'll be in New Mexico for 2 hours and Arizona for a little longer than that if you follow I-10. Then 5 hours in CA to get to LA.
Best view I had was Arizona at sunset/dusk headed west. Absolutely breathtaking. Google Arizona Highways and click Images to see what I mean.
If you plan to make diversions to see sites there may be some things of interest. I've done the run a few times in a straight shot of 22 hours on the road.
Sort of depends on where in Texas you are travelling from. I agree with @Moop that I-10 is pretty blah, as well as I-20. I typically drove it at night to avoid the monotony of west Texas. I-40 is really nice in comparison. You'll see cactus deserts, painted deserts, and mountain pines on the journey through AZ on I-40.
I lived in Central Texas and visited family in Phoenix almost every year for 12 years. I would try to get on I-10 as quickly as possible. HWY 190 has a lot of deer to watch for and you have to slow down for every little town along the way (nice though if you like checking out small towns, but the reduced speed limits get aggravating after a while). However Brady is kind of a nice little town. I think they have a small waterfall on the east side of town if I remember correctly. Things stay pretty boring until you hit Iraan where there are some very small mountains and some elevation changes.
I'd probably go on HWY 290. If the timing is right, then you can catch a great photo opportunity in Johnson City at the flower farms there. Either Johnson City or Fredricksburg allows for open alcohol on the public sidewalks (not in your car though).
If you stay at a motel along I-10, note that there is a railroad track right next to it and trains like to blast their horns from midnight to 2am. The Dragoon Mountains in AZ is another interesting area with its huge boulders (another photo op). If you go I-8, Yuma has sand dunes and is where part of Tatooine was filmed. I-10 has Blythe where you should get a Slurpee and some extra water before going across the Mohave.
If you have time, the I-40/Route 66 is a lot more scenic in my opinion.
So it looks like I'll be driving my Elise from Austin to Phoenix over Memorial Day Weekend and looking for somewhere to park it in Phoenix for a few weeks...any kind LT'ers in Phoenix available to babysit an Elise until I can fly back to pick her up?
Will probably stop in El Paso and Tucson on the way to Phoenix and would love to grab a coffee/drink/meal with any LT'ers in those cities.
When you're heading out of Phoenix to SoCal there aren't a lot of interesting options from which to choose. I've driven something similar to the above listed route, but a bit different. Phoenix to Escondido, 670 miles / 12Hrs