For the most part the park is all paved roads. There are however a few dirt roads that will take you to places. They are not really that difficult to traverse. All the same though, you still might want to leave your Ferrari at home.
Because of its size and the distances between its major sights Joshua Tree is best visitied by car. In fact it's impossible to visit without a car. As far as I know there is no public transportation oferred inside the park. The car should be in good mechanical condition especially if you're visiting in summer. Gas is not available inside the park. A few roads (for example the Geology Tour Road) are only recommended for vehicles with 4-wheel drive or with high clearance.
I don't like driving in general. In southern California we already drive too much. But driving in Joshua Tree National Park is fun. The road is well paved. It curves a little and goes up and down but not too much. Then there's the scenary... you are often surrounded by pathetic Joshua trees, dramatic rock formations, and falling rock climbers. No wonder Hollywood loves to come to the Park when they need an escape.
Warning -- the Park's road construction project has been going for 2 years (as of Oct 2003). Some major sections are currently unpaved.