The fiftieth day of the Omer is observed as the holiday of Shabuot. The Halachic authorities address the question of whether or not there is a requirement to study the laws of Shabuot thirty days before the holiday, as we are required to do before Pesah. The Bah (Rav Yoel Sirkis, Poland, 1561-1640), in Siman 429, rules that this obligation applies to Shabuot, and one must begin studying the laws of the Yom Tob a month in advance. This is also the view of the Magen Abraham (Rav Abraham Gombiner, Poland, 1635-1682). Others, however, disagree, and claim that as there are relatively few Halachot that apply on Shabuot, it suffices to begin studying the relevant laws on Rosh Hodesh Sivan. This was the position of the Gaon of Vilna (1720-1797), in Siman 429, and this is the Halacha, and thus one should begin studying the Halachot of Shabuot from Rosh Hodesh Sivan.
It is customary to immerse in a Mikveh on Ereb Shabuot. The Arizal (Rabbi Yishak Luria of Safed, 1534-1572) taught that one should immerse at least twice – once to remove the weekday "garments" from the soul, and a second time to receive the sanctity of the holiday.
The question arises in situations such as this year (5772), when Shabuot begins on Mosa’eh Shabbat, whether one may take a nap on Shabbat afternoon, before the onset of Shabuot, to prepare to remain awake that night. There is a well-established prohibition against making preparations on Shabbat for the weekday, and it should thus perhaps be forbidden to take a nap on Shabbat afternoon specifically for the purpose of preparing to stay awake that night, the night of Shabuot. Indeed, some authorities ruled that one should not verbally state when he goes to lie down on Shabbat that he does so to prepare for Shabuot night. This was the view of the Sefer Ha’hasidim (266), cited by the Magen Abraham (290). However, Hacham Bension Abba Shaul (Israel, 1923-1998), in his Or Le’sion (vol. 3, p. 195), disagrees, and maintains that one may explicitly sleep in preparation for the Shabuot night learning, since this is preparation for the sake of a Misva. Nevertheless, it is preferable to satisfy all opinions and avoid explicitly stating that one sleeps in order to be able to remain awake to learn that night.
Summary: It is proper to begin learning the laws of Shabuot from Rosh Hodesh Sivan, and it is customary to immerse in a Mikveh on Ereb Shabuot. When Shabuot begins on Mosa’eh Shabbat, one may take a nap on Shabbat afternoon in preparation for the all-night learning, but one should preferably avoid stating explicitly that he sleeps in preparation for the night.