While it may seem counterintuitive, Batman's greatest teacher was actually the Scarecrow. Bruce Wayne lost his parents when he was just a child, and the loss led him to swear to try to prevent anyone else from suffering the same fate. He should have been nothing more than another spoiled rich kid, but his life took a completely different path after that fateful night - one that led to his becoming Gotham City's greatest protector.
Bruce spent years of his life traveling the globe, learning every skill he believed he needed in order to save his city. He found the world's greatest martial artists and trained under them until he became proficient in countless forms of combat. He found Henri Ducard, "the world's greatest manhunter," and learned how to become a gifted detective from him. When Bruce Wayne returned to Gotham, he was probably one of the most gifted people in the entire city. But there was one last thing he needed to figure out in order to become Batman; he needed to learn how to handle the psychology of criminals, and how to make them afraid.
Strangely enough, according to Michael Green and Denys Cowan's Batman Confidential #9, Batman learned this from the most unlikely source - one of his own villains, the Scarecrow. Dr. Jonathan Crane was a renowned psychologist, and it seems his work on the neurobiological underpinning of reward pathways and their relation to anxiety and fear proved of real help to Wayne as he invented his Batman identity and worked out how to make criminals fear him. In fact, in the early years of his career - before Crane ever became the Scarecrow - Batman even sought the psychologist out for advice.
Ironically, Crane wasn't of much help in this particular case. Batman Confidential #9 is part of one version of the Joker's origin story, and conventional psychology isn't especially helpful in dealing with the Clown Prince of Crime. "Sounds like you have a nice juicy sociopath on your hands," Crane told the Dark Knight. "You can't quell or quash insanity. You can't understand it. You can't dose it with logic, put it in a box and label it and make the world safe again. Insanity is the very dark place." In Crane's view, when it comes to the Joker, Batman is not dealing with crime; he's simply dealing with evil. This is the point where all the training Batman has gone through, from detective work to psychology studies, becomes pretty much useless.
And, of course, in the end Jonathan Crane would ultimately fall to madness himself. He would cease practice as a psychologist, and reinvent himself as a villain who used his knowledge of fear in order to launch his own reign of terror. The current Batman arc is seeing the Scarecrow ally with the Magistrate, a fascist group that is destined to overthrow Batman in DC's Future State timeline. It is ironic, then, that he was actually one of Batman's own mentors.