Palm Springs International Airport: The sensible choice. Rent a car at the airport and drive roughly 25–35 minutes to Palm Desert via I-10 or Highway 111, traffic depending. Rideshare works but is not a cheap substitute for a car once you are there; the bus option requires transfers and is about 2 hours 20 minutes, so skip it unless saving every dollar matters.
Ontario International Airport: Useful when fares or nonstop options beat PSP. Rent a car: Palm Desert is about 80 miles away, typically 1½–2 hours via I-10 when traffic behaves, longer on Friday/Sunday getaway patterns. There is no transfer worth choosing over a rental car.
Palm Springs Amtrak station: Sunset Limited/Texas Eagle service is only three times weekly and the station is in Palm Springs, not Palm Desert. Use it only if the schedule happens to fit; arrange a rental car or rideshare for the roughly 30–40 minute onward trip. It is not a practical rail-based city break.