I live in Joshua tree all winter and it is a really fun place to hike. If you go to Nomad Ventures, the gear shop in the town of Joshua Tree right outside the park, they have a great book of "25 favorite hikes" which include a bunch of day hikes ranging from .5 miles to 20 mile out-and-backs.
The "California Riding and Hiking Trail" makes a big loop and depending on your pace might be a great 2-3 night hike. Many of the hikes terminate at mines, often these are surrounded by cool old equipment. Also, if you run late on your hikes, know that even the most obvious desert trails mysteriously disappear after sunset and it is easy to start randomly following a wash instead!
Regarding camping: It is unlikely that you will come anywhere close to getting a regular camping sight in the park that time of year, as climbers from all over come to spend the christmas/new years time in the park. However, there are "out of bounds" camping allowd, check the regulations and areas. (The desert is more fragile than it looks, but I bet you knew that!)
There is no water available in the park, so have plenty of containers with you in the car so that you don't constantly have to make the drive in and out of the park for water. :)
Don't miss the "Sams Pizza (and indian food)" or the Thai restaurant in town, if you're into that sort of thing, and enjoy!