I do this trip 3-4 times a month, going from Santa Monica to Palm Springs and back.
I-10 is the most direct route, CA 60 has about 10 miles of 4 lane twisty road through some hills where it splits off from I-10 in Banning but has a simpler route through downtown L.A. Choosing one or the other depends entirely on what the traffic is doing.
The secret to driving an RV in Los Angeles is to stay in the right hand lane with the trucks. Keep a comfortable distance between yourself and the truck in front of you (no need to tailgate!) and you'll be fine. Trucks and any vehicle towing a trailer (including RVs towing cars) are restricted to the two right-most through lanes on any multi-lane freeway, unless there's a sign saying otherwise or you're lining up for a left exit. Usually these signs are on overpasses. The right hand lane is the travel lane, the second lane from the right is the passing lane, so don't dawdle in it unless you like having trucks that want to pass breathing down your tail.
All of the L.A. freeways have speed and traffic sensors embedded in them, and you can get real time information on how the freeways are doing at:
www.sigalert.com
I check this every time I head out onto the L.A. freeways. With luck, all the freeways between where you are and where you're going will be green, and you can zip right through.
Time your trip so you go through downtown L.A. mid day (10 AM - 2 PM) and you should be fine. This means leaving the Palm Springs area about 9 - 10 AM. Or go after 8 PM at night.
Be aware if you use I-10 that you will have to exit to the right twice within a couple of miles as you approach downtown L.A. to stay on I-10 to Santa Monica. If not, you'll end up either on US 101 heading northwest out of town or on I-5 going towards Orange County and Disneyland. If you take CA 60 it's a straight shot through downtown to I-10 and Santa Monica.
There's a McDonalds just off of I-10 about 30 miles west of Palm Springs, at the 22nd St. exit in Banning. This is just before you enter the L.A. basin and is a good place to pull off and use their wi-fi to check the traffic situation. CA 60 is a Left exit about 5 miles further if you choose that route.
If you want to stay overnight in the Palm Springs area before hitting L.A., there are a couple of casinos that will let you boondock for free. One is at the Dillon Rd. exit at the eastern edge of town.