Unfortunately, the aero clubs are drying up although there are some left - particularly at some AFB's. Some bases have them at a nearby GA airport rather than on base.
Here's a link to a list of Navy Flying Clubs
http://www.cnic.navy.mil/ffr/fleet_..._programs/flying_clubs/program_directory.html
Here's the USAF Aero Club list
https://www.usafservices.com/Managers/AeroClubs/LinksDirectory.aspx
I don't know how current these lists are.
If there is an active aero club on the field you can usually fly GA & park there - particularly if you are member of an aero club somewhere (they reciprocate with other clubs). But unless it's off base at a nearby GA field, it's still on a military airfield with military regulations & restrictions, and lots of aircraft with higher priority than the aero club. You can't just get a "pop-up" clearance from the TRACON, land & park/deplane at the FBO without advance notice on a 24/7 basis (like you can at most civilian airports).
I have also heard of AD members being permitted to fly GA aircraft onto some airfields without an aero club. If you're somewhere that allows this, there's some complicated details involved - you'll need to obtain a Civil Aircraft Landing Permit (DD Form 2401) and usually have a PPR. You'll probably also need these at an aero club owning base if you're flying a private (i.e. non aero club) aircraft.
One other thing to consider is transient service. Bases with an aero club will usually have AVGAS, a GA mechanic and other "FBO-like" services. Bases without an aero club usually have no fuel or other services for GA aircraft. I've flown GA aircraft onto NFW and LRF, neither have any GA services. You have to plan in advance for the "no fuel or services" problem. And some bases don't allow GA at all, presumably because GA would disrupt the mil traffic or for AT/FP considerations.
As a former aircraft owner and as much as I still like to fly GA, I agree with the previous posters that it's expensive and impractical. It's also a problem if you have an inflexible travel schedule and bad weather. I'm sure you've thought this through, but there is some weather that can be safely negotiated by airline or military aircraft but is unsafe for GA. You know the saying: 'got time to spare? go by air..."
With all that said, the Aero club/NFC is about the best value available in GA today. Good luck.