It is disputed as to when a father becomes obligated to buy Tefillin for his son, and educate him in the Mitzvah of wearing Tefillin. Practically, the old custom of Ashkenazi Jewry is not to educate the child to wear Tefillin until he becomes Bar Mitzvah. Nevertheless, today the custom is to educate the child to wear Tefillin 2-3 months prior to his Bar Mitzvah. [The Chabad custom is for the child to wear the Tefillin to 2 months prior to his Bar Mitzvah. At first however the child is to start wearing the Tefillin without a blessing. After the passing of a few weeks, when the child now knows how to wear the Tefillin properly, a blessing may be recited.]
When is the child to be educated in Rabbeinu Tam, before or after his Bar Mitzvah?
Today the custom is to educate the child to wear Rabbeinu Tam from the same time as when he begins to wear Rashi Tefillin, which is two months before the Bar Mitzvah.
If the child will be Bar Mitzvah within two months from after Sukkos or Pesach is he to begin wearing the Tefillin before two months?
If Chol Hamoed will fall within the two months the child may begin to wear the Tefillin a week earlier.
Shavuos: Accordingly, the same would apply if two days of Shavuos falls on a weekday within the two months, the child may begin to wear Tefillin two days earlier.
 A father is obligated to buy Tefillin for his son, to educate him in Mitzvos, as soon as his son is mature enough to guard the Tefillin with purity. This means that the child is mature enough to refrain from flatulence while wearing them, and to refrain from sleeping in them, and refrain from entering with them into a bathroom, even to urinate. [Michaber 37/3; 1st opinion in Admur ibid; This is approximately the age of 9 or 10 years old-Kaf Hachaim 37/14] However there are opinions [Rama 37/3; Baal Haitur Tefillin 7] that say that this only applies to a child that is over 13 years of age but has not yet grown two hairs, while a child that is under 13 years of age is not at all to be educated to wear Tefillin, as he does not yet have the knowledge to guard it. [Admur ibid]
 Admur ibid; Rama ibid
Custom of Sefaradim: The custom of the Sefardim follows the 1st opinion mentioned above [the opinion of the Michaber] and they hence put on Tefillin with a blessing beginning from two years before the Bar Mitzvah. [Or Letziyon 2/44-47; Yabia Omer 6/3; See Kaf Hachaim 37/14]
 Admur ibid; M”A 37/4
 Sefer Haminhagim p. 4; See Igros Kodesh 11/76; 14/63; Shulchan Menachem 1/104-109
 Toras Menachem 1989 4/142 [printed in Shulchan Menachem 1/107]
 Likkutei Sichos 21/357; Igros Kodesh 23/89; Shulchan Menachem 1/36; Rav Groner related to me that the Rebbe replied this answer to many people.
 See Likkutei Dibburim 1/214 that the Rebbe Rayatz began to wear Tefillin on the 11th of Iyar [and not the 12th of Iyar] seemingly because Shavuos fell that year on Shabbos-Sunday and hence one day was lost.