So far as I know, the only gun club in the area has a long waiting list and membership is expensive.
There are multiple clubs in the area, including Arlington/Fairfax Izaak Walton (range in Centreville) and Alexandria Izaak Walton (Range in Stafford).
Arlington/Fairfax may be full up again, but you can contact either of them and ask.
There are some ranges available in the National Forests west of the area also.
You are in Virginia with full state level preemption of almost all local gun laws.
Cities and counties can pass very limited rules for hunting (like banning it completely).
Concealed carry is governed by state law, and the sub-municipalities must follow the state rules.
The Code of Virginia section is here
http://leg1.state.va.us/cgi-bin/legp504.exe?000+cod+18.2-308
Permits are 'shall issue' and the training is rather minimal.
Additional training is generally a good idea, but NOT mandated.
All Virginia lethal force law is case law.
'The Virginia Gun Owner's Guide' is a very good investment if you plan on carrying.