Jermiah: I am retired Air Force, and was a ham before, an SWL during, and a ham after my 20 years on active duty.
1. Assuming you're not stationed overseas but living here, what you do off-base is your business, unless it's a violation of the law like driving while intoxicated. If you're a ham, you are free to ham-it-up. Nobody in the military cares what color your shirt is, what your shoe size is, or whether you're a ham, and you don't have to ask your commander for permission.
2. If you live on a military installation, as I did at Ellsworth AFB in South Dakota (and elsewhere), you will need to check the base/post/camp housing regulation to see if you can have a transmitter and/or outside antenna. I read through the housing regulation at the time, and saw no prohibition on transmitters, and I don't recall any mention of an outside antenna. I was an SWL when I was in SD, and had a random-length wire antenna in the back yard. Relatively unobtrusive, and no problem.
3. If you live OCONUS in a foreign country, you should check with your base/post/camp/ship legal office and the local radio amateur licensing body to find out what operating restrictions, if any, there are. I worked in a base legal office for 20 years, and read through a lot of regulations in my time, and never came across a prohibition on operating a ham radio station--at least stateside.
4. When I was traveling on military business ("TDY," or on temporary duty), which I did a lot--to Alaska, Hawaii, Germany, Saudi Arabia, Bosnia, and throughout the States, I did a lot of SWLing while staying in billeting. I'd throw a long wire out of the window and use a general coverage receiver in my room. I had a ball, and nobody said boo.
5. Being in the military is great (at least, it's great about 98% of the time, but the other 2% might not be a lot of fun, depending on what you do, where you're stationed, and whether anybody is shooting at you). There are usually few, if any, restrictions on ham radio operators, and what you do off-base is up to you--not your CO.
73, Tony N3WAK