Joshua Tree National Park is only a one hour drive and very easy drive at that. From Palm Springs take Highway 10 W to Highway 62 until you reach Joshua Tree and follow the signs. Before you enter be sure to stop in at the Visitor Center on the right hand side of the road. Rangers are there and will help answer questions, provide maps and are happy to help with suggestions for hiking, camping, off roading, picnic sites, etc. There are also books and supplies for sale as well as a cafe next door. Bring water, wear a hat, good shoes and don't go near the Jumping Cholla!
Be sure to bring lots of water, and do not touch any plants, the cactus needles can be painful and the Jumping Cholla just needs someone close by to throw of some very fine and difficult to remove needles. Keep your pets with you, don't let them wander, the coyote pup below had to be rescued by animal control. The poor hiker below well...
Plants come in various sizes, as you can see below, doesn't matter how big or small they are nasty to deal with.
http://www.nps.gov/jotr/planyourvisit/things2do.htm