If I were you, I'd skip the most popular entrypoint to Joshua Tree National Park around Twentynine Palms and stay on I-10 to enter the park from the south. Here, you can find Cottonwood Visitor Center and a lovely 3.6-mile (one-way) trail to Lost Palms Oasis. You'll pass through dry desert chaparral studded with cacti into a beautiful shaded grove filled with palms and bird life (pack a picnic lunch!). If hiking's your thing, you should also try and visit the park in spring to see the wildflower blooms (around March/April, check ahead for more precise dates). Some lovely blooms along the trails in Queen Valley and Lost Horse Valley, as well as earlier color in the lower-elevation Pinto Valley on the park's southern border. Lastly, mountain biking is another great way to experience the park in a day. Lots of good trails -- from easy to hardcore -- though my favorites are the backcountry roads in the Queen Valley area. If you just want to drive, there's a marked "Geology Tour Road" that will enable you to experience many of the park highlights. The tours at Keys Ranch also offer an interesting glimpse into a bygone era.