Shavuous and Purim - the connection:

Rabbi Chaim Shmulevitz writes in his Sefer "Sichos Mussar", that Chazal have showed us the great power one can attain by overcoming ones natural character traits.

We find on Purim that Queen Esther overcame her natural character trait. It was this action that brought about the redemption of the Jews from the hands of Haman.  Esther conquered her own personal feelings, and was willing to demean herself by flattering Haman.

We use the day of Purim to give to others and to drink till "Ad Dlo Yadah", to break down our natural tendencies.

This breaking of ones natural tendencies is a prerequisite for one to truly understand the Torah.

Chazal relate to the day of Purim, as a day upon which the acceptance of the Torah was renewed.

The miraculous barley:
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There is a Gemarah in Megillah 16:a, which brings down that the day when Hamans downfall occurred on the 16 of Nissan, was the day that the Omer offering was brought in the Beis Hamikdash.

The first crop of barley ripens around Pesach time.  One is not to benefit from this crop, until one has brought an Omer of this harvest, as an offering to Hashem.
Rabbi Eliyahu Dessler writes that this is in order for us to see the hand of Hashem in our everyday work, otherwise one may think wrongly that it comes only from his own perspiration, and may fail to see the hand of Hashem.

It is a great hidden miracle, that the hand of Hashem is hidden in the growing of one's crop.  The same is true wth Purim.  The miracle occured through natural events, for the name of Hashem is not even mentioned once in the Megillah.  There are no coincidences in this world.

The frightening mountain:
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In Masechet Shabbos we learn that the Bnei Yisroel stood in front of Har Sinai.  Rav Dimi explains that "Hashem Kafa Aleiheim Har Ke Geges".
Hashem told the Bnei Yisroel, if you accept the Torah, good, if not, this will be your burial.
Comes along the Gemarah and continues that in the time of Purim, the Bnei Yisroel accepted the Torah from their own will, and not because they were forced to.

The Baalei Mussar explain this, that the Bnei Yisroel reached such a high level, after all the miracles on their leaving of Egypt and the splitting of the sea... They reached a level where they all felt that there was no way to live without the Torah, they felt forced to except the Torah, for they felt otherwise they would not be able to exist in this world.
This great level of understanding was as if they had no free choice to choose otherwise, for the truth was so clear to them, that they had no choice, and so they said, Naasei V'Nishma.
In a similar fashion, Rav Simcha Zisel, the "Sabba from Kelm", explains that it is said that the Angels in Heaven don't have any free will.  It doesn't mean that they can't choose, rather their understanding of Hashem is so great, so they wont even think of doing otherwise.

This is similar to a person who would not go outside without his clothing on or would not enter a burning furnace, not because he doesn't have the free choice, but rather because the danger or embarrassment is so clear to him, this in itself prevents him from choosing otherwise.

Once this great moment of the acceptance of the Torah passed, and the Bnei Yisroel entered Eretz Yisroel and started to live an every day life, they started to forget the great moment of the Acceptance of the Torah.
One could say that if they were asked to accept the Torah now, in their present state, they may not be willing.

However, during the days of Achashveiros (Purim), the Jew's of that generation were lacking this great light and understanding, for the miracles were done in a hidden manner, unlike the Redemption from Egypt.
In the time of Purim, the Bnei Yisroel had a choice, no one forced them to keep the Torah, and yet they still accepted the Torah.  This is the receiving of the Torah out of love and not out of fear.
 

Rav Aaron Lopiansky writes in his book that other miracles which the Jewish people may have witnessed, are merely  proof of Divine existence.  Here by Har Sinai, the Jewish people were able to perceive Hashem, in the most direct way as possible, this was an experience of the Divine himself.  Yet only on Purim did the Jews reaffirm their commitment to the Torah.

Davkah if we take a look at Matan Torah (receiving of the Torah at Har Sinai), it doesn't sound like the Jews were forced, rather by Purim it seems as if then the Jews were actually forced?

On Shavuous, the Jews were compelled to receive the Torah, not out of physical force or threats, rather the relevation at Har Sinai was so great, there was no room for doubt, it was impossible for the Jews not to accept the Torah.
On Purim, it was not the threat on their life which caused sincere Teshuva.  Rather it was the feeling of abandonment, with a powerful yearning for a Sinai-like encounter with the Divine.

The Torah may have been given to us on Shavuous, however the clinging to the Torah, was the result of Purim.  The Torah was presented to us on Shavuous, but it could only become integrated within our personality, by searching for it, like on Purim.

Lets remember the day of Purim this coming Shavuous and find that thin crack of light in a world full of darkness, for this is the best way to have a complete acceptance of the Torah.

Halacha:
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The following contains a few Halachot for Shavuous, there are many more than what is listed below, one can look in the Mishne Berurah or "A Book of Heritage", to increase ones knowledge about other Minhagim and Halachot.
 

Decorating with branches and such:
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There are many who have the Minhag to decorate their houses and Shul’s with trees, and branches, because the Matan Torah was on a mountain which was full of grass and trees.
Another reason is because we should remember the trees, for on this day the trees are judged, and we should Daven for them.  There is also a source which supports this Minhag, found in the Aggadah, where it is written: "Haman said to Achashveirosh, that it is the Minhag of Yisroel to put trees and grass on Shavuous."

There are those who are against this Minhag in our time, because the Goyim do a similar action on their holidays, but there are those who say that we don't have to worry in this case of "Not following their ways...", for the Goyim also place their trees on the outside of their houses as well on the inside; and the accepted Minhag in the cities of Israel, is to decorate.

One should still be careful, that when he cuts off branches from trees, he should not cut off branches from fruit trees, for one is forbidden to uproot a fruit tree, and there are those that say that this law applies even to cutting off a branch.

Maariv:
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Normally one starts Maariv a bit early on Erev Yom Tov, yet on Shavuous, there is a need to have 7 complete weeks, thus we wait till after the time of "Tzays Ha Kochavim" (The evening stars come out).
 

Special additions:
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Every Chag has the regular additions (Hallel, Musaf...), Shavuous has additional ones.

Amongst them:

1) Akdamos - This is said before the Torah reading.

It was written by a cantor from the city of Vermise, the same city where Rashi lived.  The cantor's son was killed "Al Kiddush Hashem" (while sanctifying Hashem's name), and this cantor also died "Al Kidush Hashem".  He left a Piyut written in Aramaic which praises the greatness  of Hashem.

2) Ruth - We read the Megillah of Ruth.

There are other Minhagim...

No sleep...:
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There are those who stay up all of Shavuous night learning Torah.  This is because of the Midrash which brings down that the Bnei Yisroel were sleeping the night before Matan Torah, and Moshe had to wake them up with sounds and thunder.  Thus we stay awake to be ready for Matan Torah.

Those that stay up all night, should wash their hands at "Amud Ha Shachar"(rise of morning)
and he should not continue learning until he has done this.
If one wants to continue to learn, he should hear the Birchas Ha Torah from someone who slept.  For complete laws about one who did not sleep at all during the night, what he should do, please have a look in the Mishna Berurah (O.C. 494) for there are many different instances, and each case may be different.

Eating milk products:
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There is a Minhag to eat milk products on Shavuous, one should make sure that he waits the appropriate time between eating milk and meat products.
He should also say Birchas HaMazon between the two meals, if he ate bread.
One should also clean and rinse out his mouth.

There are various reasons for this Minhag of eating milk products:
Amongst them:

a) The acceptance of the Torah, there was a lot of preparation needed to Kosher an animal, and there was no time, thus they ate milk products.

b) We know that Shavuous is called "Chag Ha Bikurim", and it says in the Torah,
"Rayshis **BIKUREI** Admacha Tavi Beis Hashem Elokeicha, Lo Tevashel **GEDI BAH-CHALEIV IMO**"
"The choicest **FIRST FRUIT** of your land shall you bring to the House of Hashem your G-d.  You shall **NOT COOK A KID IN THE MILK OF ITS MOTHER**." (Shmos 23:19). This hints to Shavuous, where we eat meat and milk, but of course, not together.

There are those who say that one should eat meat at both of the Yom Tov meals and they go against this Minhag, for we know that one is to be happy on the festival, and true Simcha comes from eating meat, in the honor of the festival.  Either way, even if a person eats a Milk meal, he should make it a festive meal with tasty food, so he would feel the Simcha.

Tachanun:
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There are those who don't say Tachanun all of the first 12 days of the month of Sivan.  Some say that in Chutz L'Aretz (outside of Eretz Yisroel), it is the first 13 days that one doesn't say Tachanun and on these days, one should not fast or say eulogies over ones dead, unless it is said over a Talmid Chacham.

(DT Adapted from: Sichos Mussar, Time Pieces, Rav Dessler, Darcei Mussar)
(Halacha Adapted from: Sefer Ha Todah, Piskei Teshuvos, Otzar Taamei HaMinhagim,  Minhagei Yeshurun)