In order of preference:
Lost Palms Oasis: A 7 mile up-and-back that begins at Cottonwood Springs. For the first 3 miles, it is a moderate hike but increases in difficulty for the last .5 as you approach a deep canyon bearing a surprising number of California Palms. You can add an additional 1.5 mile hike, the Mastadon Peak loop, which intersects with this trail.
This hike provides a wide variety terrain, views and fauna. The only thing it does not give you are Joshua Trees.
Lost Horse Mine: You can do this hike as un up-and-back to make it a 4 mile hike or do it as a loop for 6.2 miles. It’s a moderately strenuous hike that gives you some great views of the park as you climb, as well as a great visit to an old gold and silver mine, complete with tipple and mill.
Pine City: A 3.5 mile up-and-back hike to the only pine trees in the park. A moderate hike that has options to make it much longer. The trail leads to a collection of giant boulders shaded by California Juniper and Pinon Pine trees. The “city” was apparently a small collection of wooden cabins inhabited by miners who worked the nearby Lucky Boy Mine.
Barker Dam / Petroglyphs Loop: An easy hike of maybe 1.5 miles. It takes you to the small lake created by a dam built to hold water for cattle then into a boulder lined valley. There is a small side track to the “Movie Petroglyphs” named so because a movie crew enhanced ancient petroglyphs with some paint to make the filming of them easier.
Hidden Valley Nature Trail: This easy hike is only 1 mile but is a complete delight. Do not pass it up.