Kind of warm in June. The park is a transition zone that includes the Mojave (high desert) and the Colorado (low desert). Coming from the north (bay area right?), you might start at the West entrance in Joshua Tree. It's called Park Boulevard, you will gain elevation. Look at your map and note Hidden Valley area, Ryan Mtn area, Sheep Pass, Jumbo Rocks. Very special rock formations, and a climbers mecca in the winter. Lots of hiking trails in the area. Keys View is paved and the highest point you can drive. Great view of Palm
Springs and the Salton Sea, even a bit of Mexico. Do this for sunrise, not sunset unless you want your view blocked by smog. If you take Pinto Basin Road to the Cottonwood Entrance off of I-10 you will be in the low desert, different plants different scenery. Coolest temps to camp will be between Hidden Valley and Jumbo Rocks.
For off road, look at Berdoo Canyon, Pinkham Canyon Road. Old mining remains are around the Old Dale Mining District. Your rig is built right? So no worries, nothing too technical. Plenty of trip reports on these trails on Expedition Portal.
What am I missing? Hit me up.