My favorite route to Southern California: from Tulsa take I-44 to OKC, pass over I-35, the turnpike continues as the name changes to the Kilpatrick Turnpike westbound. This will let you bypass going through OKC downtown. Stay on the Kilpatrick Turnpike going west, after you go past the Quail Springs Mall area keep going. It will curve to the left, stay in it, it will then take you Southbound to I-40. Take I-40 west to Amarillo. At Amarillo take I-27 south a short ways, get off I-27 and get on US HWY 60 WB. You'll go through Hereford, cross into New Mexico. At Clovis get on US HWY 70 headed to Roswell. At Roswell keep an eye out for UFO's and extraterrestrial aliens, there is a UFO museum in Roswell. Stay on US HWY 70, go through Ruidoso to Tularosa-Alamogordo. Get on US HWY 70 to Las Cruses. At Las Cruces get on I-10 and mosey on to the AZ border to Tucson. Really cool place to see:
After you cross into AZ on I-10 in about 20-maybe-30 miles you'll go through a location called Texas Canyon, do this in the daylight so you see the AMAZING landscape of GINORMOUS boulders, it's awesome, so pull into the rest area and marvel at this, take pics.
I live in OKC, I no longer take I-40 unless I have a specific reason, I greatly prefer the scenery of what I call the Southern Route. It's much prettier and interesting and as well has better weather.
BTW if the weather is bad between OKC and Amarillo, just stay on I-44 to Wichita Falls TX. Get on US HWY 277 or 380, try to pass by South of the Lubbock area and work your way West to New Mexico, depending on weather conditions just look at your map-atlas. I try to avoid arriving at El Paso because that's too far South and too many miles, unless you just want to see that area, if weather is bad enough just head for El Paso by whatever way as necessary. If you're not in a hurry the little towns and little highways are interesting. Have fun.
Have fun, enjoy your trip, be safe.