It is customary among Sepharadim to read one chapter of Pirkeh Abot every Shabbat during the period from Pesah to Shabuot. This custom is mentioned by the Abudarham (Spain, 14th century) and by Rabbi Menahem Azarya (the "Rema," Italy, 1548-1620) in Ma’amar Hikur Din (1:19). It appears as well in the work Eretz Haim (292:23).
Various reasons have been given for this custom. One reason is that during this period we are to prepare ourselves for accepting the Torah on Shabuot, and we therefore learn Pirkeh Abot, the bulk of which contains motivational material about the importance of Torah study and Misva performance. Furthermore, this is the period when the weather becomes warmer and more pleasant, which increases the temptation of physical delights. We seek to quell this effect and restrain the Yeser Ha’ra by studying the words of Musar presented in Pirkeh Abot. A third reason is that unlearned Jews would come to the synagogue for Minha on Shabbat, and it thus became customary to read Pirkeh Abot so they could learn the Musar of our Sages.
Furthermore, the Sages famously teach that "Derech Eretz Kadema Le’Torah" (proper manners must come before Torah), and therefore since much of Pirkeh Abot is devoted to matters involving Derech Eretz, we read these Mishnayot as part of our preparation for the acceptance of the Torah on Shabuot. This reason is mentioned by the Abudarham.
Finally, this period, during which Rabbi Akiba’s students perished, is a period of "Din" (divine judgment), which naturally causes people to become more conscientious and God-fearing. As people’s hearts are more open to religious admonition during these weeks, we read Pirkeh Abot whose teachings are likely to affect us specifically in this period.
It is thus proper for everyone to follow this time-honored custom and read Pirkeh Abot every Shabbat between Pesah and Shabuot. Our practice is to observe this tradition by delivering special Derashot on the material of Pirkeh Abot during these weeks after Minha on Shabbat.