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

Omer is counted for 49 days, from the second day of Passover to the day before Shavuot.  Omer connects the two holidays both physically and spiritually.  In Biblical times, an omer (a specific measure of volume) of barley was brought as an offering to the Temple in Jerusalem on the second day of Passover.  Once the barley offering had been brought, all grain changed status from “chadash” to “yashan” and could be eaten.
Due to tragedies that occurred during the omer period, especially the death of 24,000 students of Rabbi Akiva during the second century, we observe a semi-mourning period during 33 days of the 49 days of the omer.

You may swim during sefirat ha'omer.

These are forbidden during the chosen omer period:

• Haircuts
• Shaving
• Listening to any music, even pre-recorded
• Getting married.

 

The ideal time and way to count the omer is to wait until dark, stand up, say the blessing, and then count the omer for that day.
SITUATION You want to count the omer before dark (as is often done in synagogues at the end of ma'ariv).
STATUSSince the correct practice is to count the omer after dark, you should:

• Count again with a blessing sometime after dark, or else 
• Count during the following day (but before sunset) without a blessing.

WHAT TO DO  Do the counting without saying the blessing and make a condition (mentally or verbally, either is OK) that if you remember to count the omer again after dark, the first time you counted, does not count (!) Then, if you do remember after dark, say the blessing and count the omer for that day a second time.

 

Women and girls are not required to count the omer. But if they do, they say the blessing (if appropriate) and it is a mitzva for them.

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

You may cut your nails during 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.

The only halacha of “counting the omer” is to say the blessing and then count the omer.  Anything else is custom. 
 

Situation
A boy becomes bar mitzva during the omer.  He has been counting the omer and has said the omer blessing without missing a day.
What To Do
He continues counting as he had been doing.
Note Boys should be counting the omer with a blessing from before they become bar mitzva, so there should not be any break.

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.