Some of the greatest concentration of excellent climbs are on the formations within the Lost Horse Area of Joshua Tree. Hemingway Buttress and Dairy Queen Wall are quite popular, including routes such as Overseer (5.9), White Lightning (5.7), and Frosty Cone (5.7).
Drive down the Lost Horse road to access the Rock Garden Valley, where you'll find a strenuous approach up boulder fields to the Freeway Wall (Cake Walk - 5.9), Tiny Tots Wall (Dinkey Doinks - 5.8), and the Shorter Wall with a number of excellent 7's, 8's, and 9's). Park further along the road to access the Lost Horse Wall, with several good multi-pitch climbs, the Atlantis Wall, and Mount Grosvogel -- both featuring several excellent, easier climbs.
You may want to skip climbing the Aiguille de Joshua Tree, however. Only 5.6, but you'll probably have to free-solo it (and downclimb). Aiguille translates to "needle" -- you'll understand the name when you see this unique formation.