L.A. to El Paso
1) I-10 but take the Phoenix bypass to avoid the traffic. This goes down to the I-8 just west of Phoenix and over to Tucson. Are you driving it in a single day? Its 12 hours.
2) Not much past Palm Springs until Phoenix or Tucson. Post on the Southwest boards for suggestions there.
I make this drive (and then onward to Austin or Houston) several times a year.
Starting from LA, you certainly don't want to go through Yuma, and even the Hwy 85 bypass around Phoenix to I-8 is unnecessary except during peak traffic times, especially if you want to stop for a meal within 50 miles or so of Phoenix. (For food, the bypass route is uninteresting, to say the least.) See the Southwest board for Phoenix and Tucson. Depending on your schedule, the latter is probably the best place to eat, because there is virtually nothing worth having between there and Las Cruces, NM, and only a few decent places in Las Cruces. (Anyone who can knowledgeably correct my views on this part of the trip will earn my eternal gratitude!)