It is customary — but it is not a Law — to decorate the home and synagogue with leaves and branches, but is forbidden to bring in an entire tree, no matter what size. We may not prepare these during the Holiday, but only before; if they were not cut or prepared before the Holiday, it is forbidden to do it during the Holiday.
There are a number of reasons for decorating with branches and leaves. The primary reason seems to be because on Shavuos we are judged concerning the trees, and we must remember to pray that the trees grow well and healthy.
Another reason we decorate with greens is to remember Mount Sinai, on which the Torah was given. Even though it was a mountain, it was verdant and lush with green.
Moses, who was born on the seventh day of Adar, was hidden by his mother for three months, until the sixth day of Sivan. He was then placed among the river weeds. We therefore spread greens to remember the miracle that was performed for Moses at that time.
The Midrash teaches us a parable. A king planted a garden. After some days, the king looked at the garden and found it full of thorns. He was about to destroy the garden, when he saw a rose blossoming in it. The king declared, «For the sake of that one rose, I will not destroy the garden!» So too, although the world is mired in sin and degradation, for the sake of the Torah, and for the sake of the Jewish People, whose purpose it is to fulfill the Torah, the world is not destroyed, but rather, the whole world is saved.
The first night of Shavuos it is the custom to stay awake all night (until dawn) and study Torah. Some people say Tikun Lail Shavuos, which consists of representative quotes from every section of the Torah, from the Written Torah to the Oral Torah. But many people simply study any Torah they have the ability and knowledge to learn, and most people attend a lecture or dialogue session.
The Torah reading in the synagogue for the first day of Shavuos is the Torah’s telling of the day that the Creator spoke to us at Mount Sinai, and told us of the Ten Statements (which Christians have ignorantly renamed the Ten Commandments).
During the day of Shavuos, it is customary to eat dairy foods. We eat the dairy foods, recite the after-blessings, take a half hour or an hour break, and then eat the Festival Meal, which by Law must contain some meat. (If necessary, chicken can also count as meat, but people unable to eat meat or chicken for health reasons are excused.) Some people eat challah bread twice, once with the milk and once with the meat, to commemorate the Two Loaves brought as gift offerings at the Holy Temple on Shavuos. (We may not use the same bread at each meal.)
There are a number of reasons for the eating of dairy on Shavuos. One reason is to recall that when Moses was pulled out of the water on this day, he would nurse only from a Hebrew woman and he refused the milk of any other woman. Therefore, we eat dairy foods to remember this.
Another reason is to remember that until the Children of Israel received the Torah they were permitted to eat non-kosher meat. Once they received the Torah they were not permitted to eat anything non-kosher, or even to use the dishes and utensils they had used with non-kosher food. So, after they received the Torah, they had to eat uncooked dairy foods until they could make their dishes and utensils kosher, or make new ones.
Some have the custom to eat foods with honey and milk, to recall the verse, «Honey and milk are under your tongue» (Song of Songs 4:11), which refers to one who has attained the knowledge of Torah.
The Second Day of Shavuos: Many people read the entire Book of Psalms on the second day of Shavuos, because that is the day of both the birth and death of King David.
The Torah reading in the synagogue for the second day of Shavuos is the Torah’s instructions about the various Holidays throughout the Jewish year.
During the second day we study the Book of Ruth. There are a number of reasons for this.
The events of the Book of Ruth occurred around Shavuos time, as it says, «…until the end of the barley harvest and the wheat harvest…» (Ruth 2:23)
Ruth converted and accepted all the Commandments, just as we did at Mount Sinai.
King David was born and died on Shavuos, and the Book of Ruth recounts his ancestry.
A bit deeper: The Written Torah says that a Moabite who converted to Judaism may not marry an Israelite. The Oral Torah explains that this refers only to a Moabite, not a Moabitess. A Moabitess who has converted to Judaism may marry a Jewish man. (The reason for this is not difficult to understand, but it is too involved to discuss here.) Were it not for the Oral Torah, Boaz would not have been permitted to marry Ruth. It is therefore the Oral Torah that made King David possible. The Book of Ruth was written by the Prophets to show us that the Oral Torah and the Written Torah work together as a unit, and were both given to us at Mount Sinai.
The lesson within is therefore very pertinent to Shavuos, the time we received the Torah.