First thought is to actually look at the job requirements. For the FBI, (or any GS-1811, really) there’s plenty of “careers” pages on .gov sites telling you what someone needs to get the job. With the FBI, there’s an entire site on the subject.
Off the top of my head, for an 1811, you need to be free of any serious medical or psychological disorders (bipolar, major depression, schizophrenia, ect.), you need a bachelors degree (the FBI disqualifies political science and psychology degrees, btw), a valid driver’s license, you cannot have forgotten to file your tax return (ever, so far as I can tell).
It’s going to be dull reading, but it will introduce you to concepts you probably haven’t seen before in fiction that are worth knowing about. Things like Law Enforcement Availability Pay (LEAP) might not seem important, and may not have a place in your story, but being aware of them, what they are, and then actually being able to use the terms can do wonders for building verisimilitude.
There’s also a lot of fiction and non-fiction books out there from retired members of the Bureau. In general, even the fiction from former Law Enforcement Officers will include details that someone with a civilian background is likely to gloss over.
Personally, I still like Police Procedure & Investigation by Lee Lofland, along with Forensics by D. P. Lyle, as quick primers for criminal investigations. Neither of these deal with the FBI itself, but it’s a place to start.
For characters with a military background, I’m not even sure where I’d start to suggest. There is a lot of literature on the military. From the bureaucracy, to personal experiences, to doctrine and theory. So without more information about exactly what you’re looking for, I’m not sure what to suggest there.
That said, you’re also going to find that there are a lot of ex-military in the Bureau. Someone with more hands on experience there will have to correct me if I’ve been mislead, but socially, the Bureau is a mixing point between former military, and local or state law enforcement veterans who moved up. You can have characters with a background in neither, but they’ll be far more isolated.
If you’re specifically looking for the overlap between military and law enforcement, you’re probably going to find that when you’re conducting a lit review on the FBI.
Incidentally, @skypig357 may be a useful person to poke, for this one. As I recall, Air Marshals aren’t technically GS-1811s (because they lack investigatory powers), but he has had first hand with Federal law enforcement, and can probably help you there.