Joshua Tree 1st week December

20F in December should be fine. Be aware that it can sometimes get very windy at night so try to find a spot behind something to block it off.  As for bringing a tarp, I normally cowboy camp in just a bivy, but I would verify the weather before going as it can rain starting late November. Even if it rains, a tarp is all I normally bring.

Across the northern part of the park is the California Riding and Hiking tail which crosses a road several times over it’s ~36 mile length.  The entire thing makes a great 2-3 day one way hike in the cooler months.  You can pick any section of it you like.

I’d pick up a copy of the National Geographic or the Tom Harrison Joshua Tree maps which show you all the trails and some of the easy cross country routes in the park for the more adventurous.