So What's the Problem?
From that point on in the movie, they don't teach him jack shit.
Sure, they tell him aliens exist and give him the background on the MIB in the form of a quick tour, but from the moment that J puts on the suit, K (Tommy Lee Jones) throws him into the job headfirst and just expects him to catch up or ... uh, die, we guess?
"You know what totally wouldn't result in dozens of civilian casualties? Giving an inexperienced rookie a powerful and wildly inaccurate weapon and then not telling him."
Agent J is left to figure out how to use an impractically reckless gun and how to fight a giant cockroach unarmed (and other tasks that turn out to be equally dangerous, like delivering a predatory alien baby). Any one of those tasks alone would be difficult for an expert, and J has to stumble through it without so much as a company handbook.
"You're making coffee next."