With your suggestion of Joshua Tree NP I will assume you like touring the outdoors. A few outdoorsy places worth visiting near LAS are Valley of Fire State Park, Red Rocks National Conservation Area, and the Hoover Dam. That's easily 1 1/2 to 2 days of activity.
Joshua Tree is certainly worth visiting if you like the outdoors. Its lands span high desert, low desert, and transition zones. So it's interesting from a number of natural perspectives. A road crosses the park with many short trails for hiking and photography. You can commit a full day to the park and see plenty of interesting things. You could also commit 3 days and delve a bit deeper into the various bits of backcountry. Be warned that the subtle pleasures of the area are well hidden. To the untrained eye they look like a desolate, boring desert.
In Palm Springs, look into taking the Palm Springs Aerial Tramway (
http://www.pstramway.com/) to the shoulder of Mt. San Jacinto.
If you go to Death Valley, there is plenty to see there. Again, though, it is well hidden in the form of a desert.
As you head south from Death Valley toward LA along Route 14 you will pass through Red Rock Canyon State Park. It's worth a 1/2 day visit to break up the drive.