In Naruto, during the training at the Academy in Konoha before a person becomes a genin, the work shown seems to be basic and focusing on nonspecializing skills like throwing a kunai. When they become genin, however, people seem to already know specialized skills such as Kiba or Shikamaru's abilities. When does training for this occur?
On their cases, each clan specializes their own techniques, and teaches those to their young. The training for those actually occurs after they are genin (usually). There are, of course, some exceptions. Such as Sasuke learning the Fireball technique when he was very young.
Naruto didn't learn any specialized technique before the Rasengan (sure, he had the shadow clones he learned from a forbidden scroll, but that's hardly specialized).
Basically, it happens after they graduate from the academy, sometimes slightly before.