Shavuot (Yom HaBikurim in the Torah) celebrates and commemorates the giving of the Ten Commandments to the Israelites at Mt. Sinai.
Beginning with the second night of Passover, the Israelites who left Egypt underwent 49 days of spiritual improvement and purification until they were ready to receive the Torah from God (Shavuot ends this 49-day “omer” period).  We can undergo a similar process of spiritual development each year during these 49 days (how to do that is beyond the scope of this website). According to our tradition, the Israelites in Egypt had sunk to the 49^th level of spiritual impurity (tum'a). The Israelites had to raise themselves in 49 daily stages to be worthy of receiving the Torah.  Several books and siddurs portray the 49 days of the omer as corresponding to the Seven Sefirot embedded in the seven weeks.  This awareness can help us work on and maximize the power inherent in each day of the omer to fix that particular sefira in ourselves. We thus relive this transition from slavery to freedom and the service of God each year as we try to perfect our midot (personal characteristics) to again be worthy of receiving the Torah on Shavuot.

 
Symbolism of the Shavuot Offering
In the Temple in Jerusalem, the only communal sacrifice of leavened bread was on Shavuot. Leavening in dough is compared to arrogance in humans (people puff themselves up to look more important than they actually are). During Passover we destroy, and refrain from eating, leaven--just as we try to destroy/remove arrogance from our personalities. After Passover, we continue to work on our personal traits (midot) until we reach Shavuot, when we celebrate receiving the Torah.  At Shavuot, we Jews have a right to feel important, since we are spiritually elevated by virtue of having been given the Torah.
 
Shavuot: Universal Customs
The universal custom is to eat at least one dairy food during Shavuot.
Possible reason  At the time the Israelites received the Torah, they did not have any kosher meat (they had not been required to eat kosher until then) and so the only food they were permitted to eat was dairy food.
Another universal custom is to stay awake all night (if possible) studying Torah.
 
Shavuot: Symbols
Unlike other Jewish festivals, Shavuot has no concrete symbols and no specific unique commandments/mitzvot, other than sacrifices that were brought in the Temple in Jerusalem.

You may swim during sefirat ha'omer.

You may say she'hecheyanu during sefirat ha'omer.  

You may cut your nails during sefirat ha'omer.
 

The blessing over counting the omer is “al sefirat ha'omer.”
 

You may buy and wear new clothes during sefirat ha'omer.
 

The mourning period during the counting of the omer (sefirat ha'omer) lasts for 33 days, but there are three ways to count them:

1. Second night of Passover until Lag ba'Omer (33rd day of counting the omer).
2. Rosh Chodesh Iyar until the third day before Shavuot (excludes Lag ba'Omer).
3. Second day of Iyar until Shavuot.

You may change your observance from year to year, if necessary.

Here are the times when men and women are forbidden to get haircuts and men are forbidden to shave, including their necks (from most severe restrictions to least):
Most Severe Restrictions

• Mourner for parents (no shaving for the first 30 days and no haircut until three months after his or her previous haircut).
  Note If mourning for other relatives, you may get a haircut after 30 days.
• From Saturday night before Tish'a B'Av until the evening after Tish'a B'Av.
• Chol ha'moed.

Next Most-Severe Restrictions
Shiv'asar B'Tamuz (17^th of Tamuz) until the evening after the Shabbat preceding Tish'a B'Av.

Least-Severe Restrictions
During 33 of the 49 days of counting the omer (sefirat ha'omer); that is, either from the second day of Passover until Lag Ba'Omer (33rd day of the omer) OR from Rosh Chodesh Iyar until the day before Shavuot, according to your custom.