Even though a child is exempt from all the commandments, and his father’s obligation to train him in them is not Biblical, but rather Rabbinic in nature— Torah Study is different.
Torah places an obligation on the father to teach his male child Torah, even though the child himself is not obligated to study; “And you shall teach them to your sons in order that they speak mainly about them.” (Devarim 11:19)
At what point is the father obligated to teach his son?
When the boy begins to speak, his father teaches him “The Torah that Moshe has commanded us…. (ibid 33:4) and the first verse of the section of “Hear O Israel”. Afterwards, the father teaches the boy, slowly but surely, many verses by heart, until the boy reaches the age of five years; meaning until the beginning of his fifth year.
Then the father teaches him to read the Tanach, little by little and at home, until the boy reaches the age of six or seven years. Prior to that, in the forth year, the father had to have taught him the letters of the Torah, in order that the boy be prepared to read in the Torah the fifth year.
When the boy reaches the age of six or seven complete years–it depends on the health and strength of the boy–he [the father] brings the boy to a teacher to read in the Torah the entire day. The boy would do this until he was ten years old, by which point he would have read the entire Tanach many, many, times.
In those days, people spoke Hebrew, and, [of course,] when a child would begin to speak his father would talk with him in Hebrew. Therefore, there was no need to teach the children the meaning of the words. They had only to teach them how to read the letters with proper pronunciation and with the proper tune, and also how to read those verses that are to be read differently than how they are to be written.
In those days, the vowels and notes were not written; [the text of the whole Tanach looked] like our Torah scrolls; they, therefore, needed to work for five years learning the entire scriptures many times, in order to learn the pronunciations and musical notes by heart.
Afterwards–five years learning the Mishnah–a body of work comprising the laws without reasoning–by heart.
Afterwards–five years in Talmud–[related but not identical to the Talmud]–which is [learning] to concisely know the reasoning behind the laws, as well as their, [respective], sources; is it derived from the written Torah through one of the thirteen rules of Biblical exegesis [given to Moshe at Sinai] or through other methods; or is it a tradition going back to Moshe from Sinai; or is it from logic; or is it mandated from the Sages as a fence and a wall preventing us from transgressing the words of the Torah?
Then, a man spends his entire life–each man according to his intellect and ability–in talmudic dialectic, pointing out contradictions and coming to resolutions, descending deeper and deeper into the depths of the reasons and exegesis, understanding one thing from another and innovating new halachot and exegesis.
To the aforementioned [process] our Sages referred when they said, “at forty, a man reaches understanding” (Avos).
Likewise [included in this process] are all the words of our sages and their mysteries, known as “Hagadot”, that they leaned and expounded on the verses of the Tanach.