It is one of the Islamic etiquettes to greet someone by shaking hands. There are a number of Hadith reports, which stress and recommend a Muslim to do this.
But, we sadly note that we see in many areas that people have invented a custom of shaking hands after particular prayers.This particular type of greeting is not proven through Ahadith and Juristic decisions of Islamic scholars. No doubt, shaking hands is a Sunnah and observing it is considered as an act of worship in Islam. And, its proper time is the time of meeting or departing from people. To shake hands after particular prayers or after some ceremony is un-Islamic. So, it goes against Sunnah to shake hands particularly on the Day of Eid. Therefore, the Muslim scholars have declared it to be Bid'ah to greet or shake hands after genuine times or prayers. Mulla Ali Qari says: "Some times some people meet each other and they become busy in some conversations and intellectual debates for a long time. Then when they are finished with the prayer, they start shaking hands. Therefore, our Jurists have explained that this is Makrooh and it is one of the heinous innovations of our times."
The above lines clearly clarify that the current way of shaking hands on Eid, etc, is an act of Bid'ah. It happens that the family members; the son, the father and others get out of house together and reach Eidgah. On the way to Eidgah they talk to each other. But, after finishing with the Eid prayer the same people begin to shake hands with each other as if they had not met each other before. So, this is an innovation.
In brief, one may shake hands either at the time of meeting or departing. There is no particular way of shaking hands for Eid.
In the same way it is a Sunnah to embrace someone on some particular occasions but it is a Bid'ah to hug on the time of Eid. Some other scholars like Abu Hanifah and Muhammad have absolutely prohibited Muslims from hugging one another. Imam Muhammad says, "It is undesirable (Makrooh) for a man to kissa man's face, hand or any part of his body or embrace.
However, embracing some one is permissible but it should not be on the Day of Eid in a special way as the people practice it in our present times. After meeting someone after a long time, making muaanaqah is established in the Hadith. One may make muaanaqah towards the right side or left side. However three times muaanaqah is not established.