Shavuot has a number of customs which are deeply ingrained in the celebration of the holiday. Among these are the following:
Staying up all night to learn Torah on the first night of Shavuot
Spreading greens and flowers in the synagogue on Shavuot
Eating Dairy Meals on Shavuot
Staying Up (Without NoDoz)
There is a custom among the People of Israel to stay up all night to study the Torah on the first night of Shavuot. One of the reasons given for this custom is that it is to "make up" for the behavior of a large number of Jews who were present at Sinai, at the "main event," so to speak, yet they went to sleep that night. And Hashem had to wake them up with peals of thunder and Shofar blowing, to receive the Torah.
It is said in defense of those Jews that they slept "l'shem shamayim," with good intentions, for they felt that they would be better able to absorb and withstand the experience of Hashem's Revelation, which they knew was coming in the morning.
Those who stay up all night should wash their hands in the morning as usual, but without making the "brachah," or blessing, of "Netilat Yadayim," which is made each morning when one has had a regular night's sleep. Neither should they say the regular "Birchot HaShachar," the Blessings of the Morning, which contain blessings which correspond to the various aspects of "waking up:" opening the eyes, standing up, getting dressed, etc. On Shavuot morning, they should hear these "brachot" from someone who had slept during the night, but who came to "daven," to pray, typically at an early hour, with those who had stayed up to "learn."
Spreading Greens and Flowers
There exists a beautiful custom of decorating the synagogue on Shavuot with flowers and greens, because of the vegetation on Mt. Sinai. Some have the custom of adorning the Sefer Torah with roses. That, in particular, seems to have been an ancient custom, because Haman criticized the Jewish People to Achashverosh because of their observance of that custom.
The custom once existed to bring trees into the synagogue, but the Vilna Gaon basically banned that custom on the basis of it being "chukot ha'goyim," "adopting the customs of the gentiles," who have adopted the custom of celebrating their holiday by the use of trees.
Eating Dairy Meals
There is a tasty custom of eating dairy foods on the first day of Shavuot. Some simply eat a dairy meal. Many observe this custom by beginning with dairy foods, and following it by meat, to fulfill the commandment of "And you shall rejoice," and for most people, "there is no 'rejoicing' without meat."
In this case, one has to be very careful to rinse the mouth carefully, and to wait an hour between eating the dairy and eating the meat! In the other direction, of course; that is, meat first, then dairy, rinsing the mouth and waiting one hour is not sufficient (unless one is Dutch)! Then, one must wait several hours between meat and dairy. The number of hours is determined, again, by custom. The time interval varies from six hours to three hours (German Jewish custom) to one hour (Dutch Jews).
When having dairy followed by meat in relatively close succession, one must also say Birchat HaMazon (the blessing after a meal), spread a different table cover and reset the table for meat. A hint that this is the procedure to be followed is the Shavuot-related verse, "Bring the first fruits of your Land to the House of the L-rd - Do not cook a lamb in its mother's milk" (the three-fold repetition of the latter part of the verse being the source for the prohibition of the meat and dairy combination).
There are a number of reasons offered for this custom, but, whatever the reason, it is an established custom, and as long as it is not illegal, immoral or (very) fattening, we continue to abide by it. Some of the reasons given are as follows:
The day that Moshe Rabbeinu was pulled from the water by the daughter of Pharaoh, was the Sixth of Sivan, the day on which we celebrate Shavuot. And Baby Moshe refused to nurse from a non-Jewish woman, so that Miriam, Moshe's sister, was able to get Moshe's real mother, Yocheved, to be his nurse.
Until the giving of the Torah, meat was permitted to be eaten without ritual slaughter. Once the Torah was given, all methods of killing the animal for the purpose of eating other than "shechitah," ritual slaughter, were prohibited. Since shechitah could not be done on Shabbat, and everyone agrees that the Torah was given on Shabbat, the Jews had to eat dairy.
The "gematria," sum of the numerical equivalents of the Hebrew letters making up the word, of "chalav," milk, is forty (letter "chet" (8) plus letter "lamed" (30) plus letter "beit" (2) equals forty) which corresponds to the number of days that Moshe studied the Torah with Hashem on the top of Mt. Sinai.
Mount Sinai has eight names, one of which is "gavnunim," because its appearance resembles that of cheese, "gevina," in Hebrew.
Until the giving of the Torah, the Jewish People were afraid that the milk of animals was prohibited under the category of a "limb from a living animal." This is one of the Seven Laws of Noach, which Noach transmitted to his sons, obligatory upon all of humanity, and which is the source of the prohibition of causing excessive pain to living creatures. Once the Torah was given, and "Chalav," milk, was included among the seven types of produce with which the Land of Israel is blessed, the Jewish People realized that milk was indeed permitted.