Shavuot does not have any specific Mitzvot apart from the usual Mitzvot of Yom Tov. Yet, there a number of Minhagim which distinguish Shavuot such as eating cheesecake, learning through the night, decorating the shul with greenery. Both Sephardim and Ashkenazim have the additional Minhag of saying poetic representations of the Mitzvot and piyyutim. The Ashkenazim say the Piyut called Akdamot which is said before Kiriat haTorah.
The Sephardim say the Azharot, the Admonitions before Minha. It is these Azharot and the Minhag of saying them that I want to concentrate on.
They are the list of the 613 Mitzvot listed in poetic form by Isaac ben Reuben Albargeloni (born 1043 Barcelona) which was read by Jews of North Africa, and also the Azharot by Rabbi Solomon ibn Gabirol (1021 born Malaga) and it is this version which is read by Spanish & Portuguese communities in New York, London and Amsterdam, with a poetic introduction by the 12th Century poet David Ben Eleazar Bekuda.
Originally the Azharot were read during the repetition of the Musaf. But there was halachic opposition to the interrupt of the Obligatory prayers of the Musaf, and so some read the Azharot after Musaf, and some like the Spanish & Portuguese communities of today, read them before Mincha.
In the Gola the 248 positive Mitzvot are read on the first day, and the 365 negative Mitzvot are read on the second day. The Azharot are followed by the reading of Megilat Ruth. The first two chapters on the first day, and the second two chapters on the second day. Here in Israel of course when we only have one day, all the Azharot and the Book of Ruth are read together.