You can’t do Joshua Tree and the aerial tramway in the same day from Palm Springs and do either of them any justice. It’s well over an hour’s drive from PS to the South entrance and Joshua Tree entrance, and at least an hour’s drive through the park between them without even stopping for a walk!
If you want red rocks, I seriously suggest Sedona in Arizona. It’s spectacular, and there are many, many short hikes within a very small distance, so very suitable for children. It’s only 2 hours from the Grand Canyon (but you would have to buy an Annual Pass for that these days).
On other National Parks, Zion is beautiful, but can get incredibly busy. Ditto Yosemite. Bryce is wonderful, and quieter. Joshua Tree is great too, but massive and possibly less exciting for the small people. I’ll throw in a wild card of Saguaro near Tucson which is (are, there are two halves) fabulous, with massive cacti everywhere.
If you do go to Palm Springs, which I also love, there are fabulous small canyons near Palm Desert, and a pretty good Living Desert zoo. Also, take a trip up to Idyllwild, and time your trip up from or down back into Palm Springs with the sunrise or sunset for incredible scenes.