My interest is mostly Wildflowers, so I'll concentrate on that.
The south end of the park at the I-10 entrance (Cottonwood Springs Rd.) is much lower than the north end. Flowers bloom and stop blooming considerably earlier.
Driving the road from I-10 north gives you a number of different views. There is a parking lot on the right not too far from the entrance with a nature walk. Lots of opportunity for taking flower pictures.
As you drive, watch for washes where water flows after rains. These washes will have different vegetation.
Watch for flowers as you drive along. Your chances of seeing things worth stopping for are much better at 40 mph or slower than at 55 mph. I seen vivid Apricot Mallow along
A but farther in, turn right (east) on the Cottonwood Oasis Rd. Drive to the campground or the end of the road and walk on on a trail to Cottonwood Spring.
As you drive north on Cottonwood Springs Rd., look for the Cholla (cactus) garden on the left. Sunlight more or less behind them makes them glow.
There are rock outcroppings along the road that may be of interest for you.
Other places to visit in the area:
Anza Borrego Desert State Park - to the south and west. The wildflower season there starts earlier than at Joshua Tree.
Salton Sea - south of Joshua Tree.
Amboy Crater & Mohave National Preserve - higher and later than Joshua Tree.
Antelope Valley and the California Poppy Reserve - You exit Joshua Tree at 29 Palms. Take Hwy 62 west and then Hwy 247 north. This takes you to the east end of the Antelope Valley.
There are a number of places in the valley that have great flowers in a good year.
Some resources:
http://naturalhistorywanderings.com/2015/02/21/joshua-tree-national-park-wildflower-report-feb-20/
http://www.desertusa.com/wildflo/wildupdates.html
http://theodorepayne.org/education/wildflower-hotline/
reports on wildflowers in many areas in southern and central California
reports appear every Friday from the beginning of March through May.
