The Shulhan Aruch, in the beginning of the laws of Pesach (Orah Haim 429), cites the Gemara’s instruction that we are to "learn and teach the laws of the holiday thirty days before the holiday." At first glance, this appears to require us all to study the laws of every Yom Tob thirty days before the Yom Tob. However, the Ran (Rabbenu Nissim of Gerona, 1320-1376) explains this Halacha as applying to a Rabbi who is approached with a question relevant to the upcoming holiday and a question on a different topic. If this occurs within thirty days before the Yom Tob, the Rabbi should give precedence to the question relevant to the Yom Tob, due to the imminent practical application of the question.
As for a requirement to study the laws of Yom Tob, the Ran notes the Gemara’s comment at the end of Masechet Megilla that Moshe Rabbenu established a requirement to study the laws of each Yom Tob on the Yom Tob. It is thus only on the Yom Tob itself that one must study the laws of the Yom Tob. Thirty days prior, there is no requirement to study the laws, but a Rabbi should give precedence to these laws over other material. This is the view accepted by the Shulhan Aruch, and Hacham Ovadia Yosef (Yabia Omer, vol. 2) notes that this is also the view of the Rashba, the Ramban, and numerous other earlier authorities. Therefore, there is no strict halachic requirement to study the laws of Pesach in the weeks before Pesach, though there is a requirement to spend some time learning the laws of Pesach on Pesach itself. Of course, Rabbis must review the laws of each holiday in advance of the holiday as they will be fielding practical questions as people prepare for Yom Tob.
Tahanunim are omitted from the prayer service throughout the entire month of Nissan. On the day after Pesach ("Isru Hag"), one should eat fine food and wine, as it is a quasi-holiday. This is noted by the Rama (Rav Moshe Isserles of Cracow, 1530-1572), in Siman 429. Although public fasts are not declared during the month of Nissan, the Shulhan Aruch rules that one may observe a private fast – such as for a Yahrtzeit – during this month.
Some people have the custom to fast on the day before Rosh Hodesh (which is called "Yom Kippur Katan"), and it is proper even for those who do not normally observe this custom to fast on the day before Rosh Hodesh Nissan. It is also customary to visit graves of Sadikim on this day.
The Rama brings a custom to omit "Mizmor Le’toda" from the prayer service on Ereb Pesach, but the practice of the Sepharadim is to recite this Psalm as usual on Ereb Pesach.
Summary: There is a requirement to spend time every holiday studying the laws of that holiday. Within thirty days before the holiday, a Rabbi who is asked several questions should give priority to questions related to the upcoming holiday. Tahanunim are omitted from the prayer service throughout the month of Nissan, and public fasts are not proclaimed during this month. On the day before Rosh Hodesh Nissan, many have the custom to fast and to visit the graves of Sadikim. Some omit "Mizmor Le’toda" from the prayer service on Ereb Pesach, but the custom among Sepharadim is to recite it as usual on this day.
