Well OP, you're asking this question on a board aimed at aviators....my wife is an agent and active-duty military, and I'd say most of these posts are correct about choosing aviation, but INCORRECT about the FBI. We spoke it about it over breakfast, and she's looking over my shoulder as I write this.
The military is awesome; it's a right of passage and despite the sh%$ talking between communities or services, you'll always have a bond with those who been there and done that. There's also a certain comradery when you pack up your stuff, leave your loved ones, and go live with a bunch of dudes (gender neutral use of the word) for six to eight months while crammed into a sleek grey merchant of death. All that stuff mentioned is true.
In the FBI, you get almost all the benefits of the military without so many of the downsides. When people say "be careful, FBI isn't like movies..." they're right, just like I imagine being an aviator isn't just like TOP GUN: Maverick; In the FBI you're given a gun, a car and good (not great, but good) technology and basically allowed to go run your own cases. As long as you get results, and barring a very-rare-jerk-supervisor, you'll be able conduct them however you see fit. For the most part you get weekends and nights off, but if you work them you're usually compensated with extra pay or another day off. The guy working the boring cyber job mentioned a few posts above? He's probably working really interesting casework and gets sick of people asking about the FBI, but even if he hates it, it's fairly easy to switch squads and start working gangs, or counterintelligence, or terrorism or anything else. You can reinvent yourself every few years if you like. Or let's say you don't love that cyber job, but you want the training for a post bureau career, there a million collateral duties you can do. These aren't "collateral duties" like Legal Officer or Bull Ensign, but things like SWAT, Evidence Response Team, Crisis Negotiation, firearms instructor, Undercover Employee, pilot and more. My wife thinks she's got the best job in the world, but often downplays it at cocktail parties or in her reserve unit. We have homesteaded in one city for almost 20 years, but if you want to move around or get to someplace cool you can certainly do that. Later in your career you can be stationed abroad at an embassy as part of the legal attache program.
If you go in the military first, the FBI will *probably* be there when you get out. The FBI likes to hire people with a little life experience, so if you're competitive now you'll likely be much more so after serving in the Navy. That said, in the 90s there were several hiring freezes, where people waited years to join, at which point the backlog made it much more competitive. It's been 20 years since the post 9/11 hiring boom, so that may happen again.
Military and cops have sort a special connection within the FBI, and my wife jokes that their main critical skill is being conditioned to deal with government BS; again, FBI-government BS is waaaaaaaaaaaay less than the Navy's BS. Most former-military people really enjoy being able to live outside of fleet concentration areas, have non-military friends, and be able to make their kids soccer games fairly consistently.
My wife did say that sometimes cops get frustrated with the job; it isn't going out patrolling looking to take a bite out of crime every day, but typically long-term investigations. That said, the guys she's thinking of have always ended up finding something that scratches that itch like full time SWAT, fugitive task force, or stuff like that. In 15+ years she can think of only three agents that have quit because they didn't like the job.
Going Navy first and then FBI: you'll take a slight pay cut initially, but after a few years you'll surpass your military pay. Your military time will count toward your annual leave; you'll probably start at 6hrs of leave every two weeks, then when you get to 15 years of military+FBI you'll get 8 hours. Leave on the civilian side is much more "friendly" however. It's almost never denied and when you don't have to use it day-for-day or over weekends, it goes further. She typically donates leave because she can't use it all. You can also buy back your military time, which essentially at retirement gets you 1% of your pay for every year of military service. As stated, the age limit for entry is 37 (so you can retire by the mandatory 57). During GWOT they allowed your deployed time in the military not to count, so you could get in a few years past 37 or something like that, but I'm not sure you can count on that option down the line.
Prior to 9/11 you couldn't be in the reserves, but the FBI relented and allowed it. My wife really enjoyed being able to put on the uniform, work with good people, and travel cool places....but the admin BS and micromanaging of the reserves killed her soul. Things like an exercise finishing three days early but not being able to fly home early without a mountain of paperwork, or endless reserve calls/email during the workday or family time in the evenings. Plus drill weekends were basically spent standing in line, submitting paperwork that had already been submitted and generally coming away frustrated. Her active duty time counts toward her FBI retirement though, *and* it counts toward her reserve retirement.
That was a long way of me saying that all three are great jobs, so search around, get lots of info, but know that you can't really make a bad choice here.
