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Why do we begin learning Pirkei Avos after Pesach?
After leaving Egypt, the Jewish people approached Moshe Rabbeinu, insisting that he fulfill his promise of giving them the Torah. Moshe explained that it was impossible for them to receive the Torah until they had prepared themselves spiritually for such an event. So, over the next six weeks, from Pesach to Shavuos, the Jewish people worked on improving their character traits, arriving at Shavuos spiritually refined and able to receive the Torah.

We emulate our forefathers’ behavior by studying Avos at this time of year. On each of the six weeks between Pesach and Shavuos, we read one chapter of Avos in the hope of improving ourselves and reaching Shavuos spiritually uplifted and ready to receive the Torah anew.

Why is Pirkei Avos studied after Mincha on Shabbos afternoon?
Tradition relates that this was the time when Moshe Rabbeinu passed away. Since there is a custom to recount the merits of a person after he dies, we read Avos at this time because it describes how Moshe received the Torah from Sinai.

Why is this tractate called Avos (which literally means “fathers”)?
The Sages mentioned in Pirkei Avos are our spiritual “fathers.” Just as a biological father brings a child physically into this world, the Sages mentioned in the Mishnayos bring us spiritually into the World to Come through their ethical instructions and moral teachings.
Another possible reason for the name Avos is that it is the tractate each father must study in order to know which good character traits he must try to instill in his children.

The Mishnayos mention teachings that each Sage “used to say,” but the Sages, who were the leaders of their generation, must have taught many, many things throughout their lives. What is significant about these particular teachings that warrant their inclusion?
While the Sages taught many lessons throughout their lives, only the statements which they repeated on many occasions, and which they excelled in through their personal example are recorded in Pirkei Avos.