When does specialization in training begin?
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.