The Kaf Ha’haim (Rav Yaakob Haim Sofer, Baghdad-Israel, 1870-1939), in Siman 494 (Se’if Katan 32; listen to audio recording for precise citation), discusses the custom among Sepharadim to read on Shabuot the Azaharot, which is a poem written by Rabbi Shelomo Ibn Gabirol that lists all 613 Biblical commands. The custom in our synagogue is to read the first three and last three paragraphs of the Azharot in the synagogue, as a reminder to the congregants that they should recite the complete poem at home. (We do not read the entire poem so as not to unduly extend the prayer service.) This is done before the recitation of Ashreh at Musaf. The Azharot are read on both days of Shabuot.
The Kaf Ha’haim adds that it is worthwhile to study on Shabuot the verses from the Book of Vayikra (in Parashat Emor) that discuss the Korban Sheteh Ha’lehem, the special offering that was brought on Shabuot. It is appropriate to study on each holiday topics relevant to that holiday, and thus on Shabuot, there is value in studying the subject of the Korban Sheteh Ha’lehem.
Furthermore, the Kaf Ha’haim writes, it is proper to study on Shabuot the Book of Tehillim, which was composed by King David, who passed away on Shabuot. It is customary to recite Tehillim in King David’s memory on Shabuot, and this occasion is an especially auspicious time for one’s recitation of Tehillim to be lovingly accepted by God. The Kaf Ha’haim notes that when one recites the introductory Yehi Rason prayer before reciting Tehillim on Shabuot, he should omit the passages that pray for forgiveness for our sins, as we do not offer prayers for forgiveness on Yom Tob. He also notes that when reciting the passage in this prayer in which we pray for long life ("Ve’nizke Ve’nihye…"), one should not recite the text praying for seventy or eighty years of life. Since many people live beyond eighty years, we should not be praying for only this length of life. Therefore, it is proper to pray generally for long life, without specifying a particular duration.
The Kaf Ha’haim also writes that one should try over the course of his Torah learning on Shabuot to conceive of at least one Hiddush (new Torah insight). Since Shabuot begins the new year of Torah study, thinking of a Hiddush during Shabuot is a favorable omen for success in Torah throughout the coming year. If one is unable to arrive at a Hiddush of his own, he should at least study new material or a new insight which is a "Hiddush" for him.
The Kaf Ha’haim writes that Rabbi Moshe Cordovero (1522-1570) would study Kabbalah on Shabuot, which would bring him success in his learning.
In this context the Kaf Ha’haim emphasizes that one should exert himself in Torah study on both days of Shabuot, and not only on the first day. He notes that according to one view among the Tanna’im (the view of Rabbi Yossi), the Torah was given on the 7th of Sivan (the second day of Shabuot), and not on the 6th, and it is therefore important to immerse oneself in Torah learning even on the second day of the holiday.
There is a well-known custom to partake of dairy products on Shabuot. Among the many different reasons given for this custom is an explanation suggested by the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), who noted that the numerical value of "Halab" ("milk") is forty. We eat on Shabuot foods made from milk to commemorate the forty days that Moshe spent atop Mount Sinai receiving the Torah. Additionally, the names of the three letters that form the word "Halab" are "Het," "Lamed" and "Bet." The "inner letters" of these three names (meaning, the letters after the first letter) are "Yod" and "Tav" (from "Het"), "Mem" and "Dalet" (from "Lamed"), and again "Yod" and "Tav" (from "Bet’). These letters spell the word "Temidit," which means "constant" or "consistent." We eat dairy products on Shabuot to remind ourselves of the concept of "Temidit," that our devotion to Torah must be constant and consistent. We cannot study only on some days but not others, when we happen to feel like it. This devotion must continue day in, and day out. The Torah says, "Zot Torat Ha’ola" (literally, "This is the law concerning the burnt offering"), alluding to an association between Torah and the daily Ola offering brought in the Bet Ha’mikdash. Torah, too, must be "daily," a constant part of our lives, each and every day.
Summary: It is customary among Sepharadim to read Rabbi Shelomo Ibn Gabirol’s Azharot both days of Shabuot. It is proper to study on Shabuot the section of the Torah dealing with the special Shabuot sacrifice, and to recite Tehillim in memory of King David. One should try over the course of Shabuot to come up with a novel Torah insight, or to at least to learn something new in Torah that he had not known previously.