Creators Kane and Finger started the Batman series, and only addressed his background after about 6 months (from Batman - The Complete History: The Life and Times of the Dark Knight by Les Daniels:
Later editions expanded the mythology. According to the DC Comics wiki, in the Earth-One series:
As Bruce became a teenager, he dedicated himself towards learning
everything he could about police procedure. He decided to track down
the city’s most famous private detective, Harvey Harris. Donning a
bright red, yellow and green costume to conceal his identity, Bruce
tracked Harris down, only to find a criminal waiting to ambush him.
Bruce used a large advertising display piece to waylay the thug,
saving Harvey's life. Harvey was so impressed with the teen's verve
that he agreed to take him under his wing. Thinking the masked teenage
boy looked as colorful as a robin redbreast in his costume, Harvey
decided to call him Robin.
As Robin, Bruce spent several weeks training under Harvey's tutelage.
Harris instructed him on the finer points of criminal detection, and
trained him in gymnastics, and hand-to-hand combat. Working together,
the two became inseparable and in the span of just a few years, they
managed to deal a marked blow against the criminal underworld.
When it was time for Bruce to enter college, he parted company with
Harvey and threw himself into his studies. He specialized not only in
criminology, but in the sciences as well. After four intense years, he
was nearly ready to fulfill his lifelong dream of becoming a
Upon graduation however, Bruce began to re-evaluate his future. He
recognized the fact that law enforcement agencies were often the
victims of an endless bureaucracy that hampered them from meting out
true justice. He could not in good conscious follow such an
But in the later series:
He spent his youth traveling the world, training himself to
intellectual and physical perfection and learning a variety of
crime-fighting skills, including chemistry, criminology, forensics,
martial arts, gymnastics, disguise, and escape artistry.
At age 14, Bruce Wayne began his global sojourn, attending courses at
Cambridge, the Sorbonne, and other European universities. However, he
never stayed long and would often drop out after one semester. Beyond
academia, Wayne successfully acquired various "practical" skills.
While abroad, he studied and received training in multiple martial
arts under various instructors and in different countries, man-hunting
under Frenchman Henri Ducard, stealth and reconnaissance under the
Japanese ninja Kirigi and other certified shinobi, hunting under the
African Bushman (the Ghost Tribes of the Ten-Eyed Brotherhood, among
others), traditional healing disciplines under Nepalese monks and even
ventriloquism under skilled practitioners. His knowledge of so many
varied disciplines has made Wayne an unconventional and unpredictable
individual. At age 20, he attempted to join the FBI, but after
learning about its regulations and conduct, Wayne deducted that he
would never be able to completely oppose crime while working within
the legal system.