The opening chapter of Pirkei Avot establishes a chain of transmission of the Oral Torah from Sinai through Moshe, Yehoshua, the Elders, the Prophets, and the Men of the Great Assembly, whose own teaching was to be deliberate in judgment, raise many students, and make a fence for the Torah (Pirkei Avot 1:1).
From Shimon the Righteous, one of the last of the Men of the Great Assembly, the chapter records that the world stands on three things — Torah, the Temple service, and acts of loving-kindness (Pirkei Avot 1:2) — while his successor Antigonus Ish Socho teaches that service of the Divine should be done without expectation of reward, out of fear of Heaven (Pirkei Avot 1:3).
The chapter then proceeds through the zugot, the paired transmitters of the tradition, each pair offering guidance in character development and communal life: Yosei ben Yoezer and Yosei ben Yochanan on attachment to sages and the home (Pirkei Avot 1:4, Pirkei Avot 1:5), Yehoshua ben Perachyah and Nittai HaArbeli on acquiring a teacher, a friend, judging favorably, and avoiding bad company (Pirkei Avot 1:6, Pirkei Avot 1:7), Yehuda ben Tabbai and Shimon ben Shetach on the conduct of judges and examination of witnesses (Pirkei Avot 1:8, Pirkei Avot 1:9), and Shemaya and Avtalyon on loving work, avoiding power, and the responsibility of sages' words (Pirkei Avot 1:10, Pirkei Avot 1:11).
The chapter culminates in the teachings of Hillel and Shammai and their descendants: Hillel's call to love peace, pursue peace, and draw people near to Torah (Pirkei Avot 1:12), his aphorisms on name, growth, and self-reliance including the famous "if not now, when" (Pirkei Avot 1:13, Pirkei Avot 1:14), Shammai's teaching to fix Torah study as a constant practice and receive others pleasantly (Pirkei Avot 1:15), and the sayings of Rabban Gamliel, his son Shimon, and Rabban Shimon ben Gamliel on doubt, silence, action over study, and the three pillars of justice, truth, and peace (Pirkei Avot 1:16, Pirkei Avot 1:17, Pirkei Avot 1:18).
The Rambam situates the broader obligation underlying these transmitted teachings, ruling that just as a father must teach his son, so too must he teach his grandson, and that every sage in Israel is obligated to teach all students even if they are not his own children, since the Torah's language of "your children" was understood by tradition to include one's disciples (Mishneh Torah, Torah Study 1:1-2).
Turning to practical custom surrounding the recitation of this chapter, Nitei Gavriel (Pesach vol. 3, perek 41) addresses when and why Pirkei Avot is recited, the differing schedules kept by various communities, its customary place before or during Seudah Shelishit together with related concluding passages and Kaddish, and exceptions to the practice such as periods of mourning or when Shavuot coincides with Shabbat.