Can I suggest you wait to make hotel reservations? The issue is that Joshua Tree is big. Very Big. The northern entrance at Twentynine Palms is fifty miles from Palm Springs. Depending on the location of the wedding, you might be better off staying there or Yucca Valley, if proximity is important to you. However, those are pretty small towns (Twentynine Palms is the biggest at 25K people) and the lodging is generally motels with a handful of national chains and not much in the way of touristic interest. Don't get me wrong, I adore Joshua Tree, it's just I wouldn't be spending my time in the cities when there are trails to hike and boulders to climb.
Palm Springs has a wide variety of lodging and restaurant choices, art galleries, etc. The adjoining towns of Cathedral City, Rancho Mirage, and Palm Desert are pretty sleepy but convenient to Palm Springs if that's where everyone is staying. We stayed in Cathedral City and it was suburban but nice. Lots of golf courses.
SNA would be a distant second place to ONT - Ontario Airport. ONT has lots of Southwest flights and saves you from having to drive 25 miles across the LA basin. LAX is a nightmare, frequently delayed and even further from Palm Springs. Burbank and Long Beach are both smaller airports that are better than LAX. However, sometimes LAX just is the cheapest by far. I put my own mother through LAX to Cleveland, after all...

Just be aware that there is no such thing as "rush hour" in the LA area. It's bad from 5am-10am, and then again from 3pm-8pm, so take that into account when planning your flight.
Unfortunately, there's not much in the way of alternative transport. I got a quote of $200 one way for a Palm Springs-LAX shuttle, and the train only runs once a day. You'll be driving for sure.
Walt