Appropriate Shoes / Clothing, Joshua Tree National Park
I visited in January and it was pretty chilly and cold in the morning. Keep in mind you are up in the mountains too. Warm weather clothes are a very good idea. Remember what your mother said about layers. ;-)
Good shoes are always a good idea. Especially here where you do a lot of hiking and walking around.
Toiletries and Medical Supplies: The only places you would be able to buy things at are in 29 Palms or Joshua Tree which is a bit of a drive out of the park. It's not very convenient, so I'd say grab all you need before you enter the park.
Photo Equipment: Wide angle lenses are especially useful here.
Camping/Beach/Outdoor Gear: Take whatever you want to camp with. You will be able to use it.
The desert climate can be severe. In the summer, it is very hot. In the winter, it is very cold. Even the milder seasons can still bring extreme temperature variation. You'll want to bring lots of layers and prepare for anything.
Even in the summer, bring along jackets or blankets. The temperatures will get quite cool at night.
Year round, you'll want a hat to keep the sun off your face. I also recommend sunglasses.
WEAR COMFORTABLE SHOES!!! Closed toe hiking or athletic shoes are really the only choice in the desert.