The Palm Springs Tram goes up to Mount San Jacinto State Park, and it is very beautiful there. The state park is a wilderness area. There is a look-out station at the top, as well as a restaurant. And there are hiking trails in the park.
Indian Canyons has several hiking trails--some short, some long. So you can choose what ever suits you best for your abilities or for how much time you have to spend there. It has been a while since I was there, but if I remember correctly, one of the trails is called the "San Andreas Fault Trail" or something like that, and does follow the fault line.
Joshua Tree National Park is a few hours drive away from Palm Springs, but I feel it is worth it. I think if you are from Alabama, you will enjoy it, simply because it is so very different from what you are probably used to seeing. I am from Pennsylvania, and my family and I enjoyed it and spent an entire day there. The park really wasn't that huge, but we found lots of hiking to do and plenty of things to see.
But as you can tell, we like to go places where we can hike. But even if that is not your thing, these three places are beautiful, and you don't have to like to hike to enjoy them.
I hope you have an enjoyable time there!