The day immediately following Pesach, Shavuot or Sukkot is referred to as "Isru Chag" and is observed as a quasi-festival. Some people make a point of wearing their Shabbat clothing and conducting a special festive meal on Isru Chag. Rabbi Chayim Palachi (Turkey, 1788-1868), in his work Mo'ed Le'kol Chai (8:43), writes that Isru Chag Shavuot is a particularly significant day, more so than Isru Chag Pesach and Sukkot, and must therefore be observed with special festivity. He goes so far as to assert that Isru Chag Shavuot is of such importance that in principle it should warrant "Issur Melacha" a prohibition against work similar to Shabbat and Yom Tov. In practice, of course, Halacha permits Melacha on Isru Chag Shavuot, but it nevertheless should be observed as a festive day, even more so than Isru Chag of the other festivals.
There is a debate among the authorities as to when we resume the recitation of Tachanun during Sivan. We of course omit Tachanun on the first of Sivan, in honor of Rosh Chodesh, as well as on the second day of Sivan, which is when God declared that we will become "a kingdom of Kohanim and a sacred nation" (Shemot 19:6). The next three days the third, fourth and fifth of Sivan are known as the "Yemei Hagbala," the days of preparation for the giving of the Torah, a festive period in which we omit Tachanun. We do not recite Tachanun on Shavuot the sixth and seventh of Sivan or on Isru Chag Shavuot, the eighth of Sivan. During the times of the Mikdash, those who were unable to bring their sacrifices on the day of Shavuot itself which in Israel is celebrated only on the sixth of Sivan - were allowed to do so during the six days following Shavuot, through the twelfth of Sivan. In commemoration, we omit Tachanun during this period, as well. According to some authorities, however, since in the Diaspora Shavuot is observed on both the sixth and seventh of Sivan, we begin the six-day period on the eighth of Sivan, rather than the seventh, and thus Tachanun is omitted on the thirteenth, as well.
For the Syrian Jewish community, this debate bears no practice relevance, as Syrian Jews in any event observe the thirteenth of Sivan as a quasi-festival to celebrate the "Musan" miracle when the Jews of Halab were saved from a planned pogrom. Syrian communities therefore omit Tachanun on the thirteenth of Sivan regardless of the aforementioned debate. Many other communities, however, have the practice to resume the recitation of Tachanun on the thirteenth of Sivan.
Later in this chapter in Mo'ed Le'kol Chai (Halacha 48), Rav Chayim Palachi writes that on the fifteenth of Sivan, it is proper for the Chazan to read aloud after the morning service Yaakov's blessing to Yehuda (Bereishit 49:8-12) and Parashat Ve'zot Ha'beracha. Reading these sections on this day, he writes, has the capacity to hasten the arrival of the final redemption.
Summary: Isru Chag Shavuot (the day following Shavuot) must be observed as a festive day, even more so that the day following other festivals. Tachanun is omitted from Rosh Chodesh Sivan through the twelfth or thirteenth of Sivan, depending on communal custom. It is proper on the fifteenth of Sivan for the Chazan to read after the morning prayer Yaakov's blessing to Yehuda and Parashat Ve'zot Ha'beracha.
