As long as all the flights are UAX, the bags you "check" at the departure gate will be returned to you at the arrival gate. If any of the flights are mainline, they will often courtesy gate check to the "final destination" and you will pick it up at baggage claim. You might be able to get the GA to courtesy check the bag thru to PSP, but that's not what typically happens IME.

All of this presupposes that you do not have a dreaded Basic Economy (N class) ticket. If that's the case, you will be required to check the bag (for a fee if not elite) at the check-in counter OR pay an additional fee to have it checked (to your final destination) at the gate.