Hi,
The Route you take would depend upon how much time you have and what else you'd like to see. I would recommend a couple of other stops, time permitting.
From LA take I-10 East towards Riverside/San Bernadino. Then hop onto I-15 North to Vegas. From Vegas take State Route 93 to drive across the Hoover Dam. 93 will take you down to Kingman to I-40. If you aren't interested in seeing Vegas and Hoover Dam, you can skip that detour and get onto I-40 directly from I-15.
Vegas http://www.usatourist.com/english/places/lasvegas/index.html
Hoover Dam http://www.usatourist.com/english/places/nevada/hooverdam.html
SR 93 and I-40 intersect at Kingman Arizona. At Kingman you have the option to head north to Grand Canyon West. This is where the Hualapai Nation has built the Grand Canyon Skywalk - a glass walkway that will allow you to "walk on air" over the Canyon. The Hualapai is a Native American tribe and the Skywalk resides on their Reservation, it is not a part of the National Park. Directions and more information is here: http://www.destinationgrandcanyon.com/map.html
Grand Canyon West http://www.usatourist.com/english/destinations/arizona/GrandCanyon/GrandCanyon-WestRim.html
Just east of Kingman, old Route 66 branches off of the Interstate. Many of our readers are interested in driving 66. And we repeatedly tell them that very little of it exists. This section in Arizona is one of the few preserved miles of the highway. It is the section we recommend people do drive if they are interested in experiencing the history of this road. Route 66 meets back up with I-40 at Seligman, AZ. Whether you take 66 or stay on 40, head east to Williams AZ. There you can take Route 64 north to the main entrance to the Grand Canyon National Park. This entrance is at the South Rim and the most visited area of the park.
Route 66 http://www.usatourist.com/english/destinations/arizona/Route66-Main.html
Grand Canyon NP http://www.usatourist.com/english/destinations/arizona/grandcanyon/GrandCanyon-Main.html
From the Grand Canyon, backtrack south on Route 64 to I-40. Take I-40 East to Flagstaff. From there take 89A south to Sedona. From Sedona take 179 South to get onto I-17 South. If you are not interested in seeing Sedona, you can get onto I-17 directly from Flagstaff.
Sedona http://www.usatourist.com/english/destinations/arizona/sedona/Sedona-Main.html
Continue following I-17 South to Phoenix. In Phoenix follow the signs to get onto I-10 East to Tucson. Near Tucson Arizona is the Saguaro National Park. Saguaro are the giant cactus' that most of us are familiar with.
Phoenix http://www.usatourist.com/english/destinations/arizona/phoenix/Phoenix-Main.html
Tucson http://www.usatourist.com/english/destinations/arizona/tucson/Tucson-Main.html
Saguaro NP http://www.nps.gov/sagu
From Tucson continue on I-10 East into New Mexico. At Las Cruces I-10 will join with I-25 and head south to your final destination, El Paso.
I hope this helps. I've included a lot of extra stops. But since they are all short detours or stops on the route, you can bypass or not stop at whatever won't fit into your schedule.
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