Edwards is a fantastic place to be. WIth Plant 42 and Mojave Airport nearby, you are right in the middle of the greatest concentration of aviation history in the world. "The Right Stuff" country. The bad part is that most of it is not well preserved or marked. But if you ask the old timers and poke around in your free time, you can dig up some really cool history.
I am sorry to say that there isn't much climbing nearby. I grew up there, and spent my childhood dreaming of going elsewhere to climb. 90% of the housing will be in the Lancaster-Palmdale metroplex. Most of the services will be there as well. But that's all desert floor. If you want to be close to the mountains, you will have to look at some of the smaller outlying towns like Littlerock and Tehachapi.
Littlerock is up against the San Gabriel Mountains and there are lots of places to go up there. Strictly "hill walking" though, nothing technical. The Tehachapi Mountains on the other side of the Antelope Valley are likewise nontechnical. There is no technical rock to speak of anywhere in the vicinity.
If you are single, your best bet for climbing would be to find a place to rent in Mojave (slim pickings). You can leave your gear bag packed by the door and swing by after work on Friday and head up to the the sierras. If you have a wife and/or kids, I would look for something on the west side of Palmdale or Lancaster close to the 14. Then you can make your escape in any direction - north to the Sierras and Inyo's, south to the San Gabriel's, west to the coast range and Sespe Wilderness or east out the 138 to J-Tree.
Just a bit north (45-60 minutes maybe?), and west, you can get into the upper Kern River area, and up into Domeland Wilderness, Kennedy Meadows, etc. 90 minutes north or so and you're in Alabama Hills.
Sadly, anywhere in the Antelope Valley sucks. It's the High Desert and that means 100+ F. everyday from March though October, then the rest of the year it's freezing and windy as hell...plus it's redneck and tweakerville. Your only hope is Wrightwood, but it will be a bit of a commute...worth it though, considering the other choices. Anywhere else nice, is going to be at least an hour drive from Edwards.
Nearby climbing includes Devils Punchbowl, ACH Tunnel Crag, Spring Crag, little further away is New Jack City, Big Bear areas, and Joshua Tree is less than 2 hrs away...Red Rocks under 4hrs!