There are more than two dozen hikes you can choose from in the park. Check out the full list
here. I suggest bunching a few together back to back.
In the one morning I did the Barker Dam Loop and Hidden Valley routes. Then I did the Ryan Mountain, Skull Rock and Split Rock hikes later the same day.
Walk 1 - Barker Dam Loop and Hidden Valley 3.5km/2 hoursThey are only a few km's each and give a good overview of the "Joshua Tree end of the park". Barker Dam is a loop track and showcases some of the cultural history of the early cattle ranchers.
Watch for bighorn sheep. Hidden Valley loops around a lovely boulder-strewn valley.
Walk 2 - Climb Ryan Mountain5km/1.5 hours
It's about 5km up and back. Not overly steep. At the top you can see the whole park and through to Palm Springs.
Walk 3 - Skull and Split Rocks2.5km each /1.5 hours each
Heading off from Skull Rock carpark you can scoot around via several tracks and around Split Rock as well, a total of about 7km.