The Sunnahs that the Muslim should observe on the day of Eid are as follows:
1 – Doing ghusl before going out to the prayer.
It was narrated in a saheeh hadeeth in al-Muwatta and elsewhere that Abd-Allaah ibn Umar used to do ghusl on the day of al-Fitr before going out to the prayer-place in the morning. Al-Muwatta 428.
Al- Nawawi (may Allaah have mercy on him) said that the Muslims were unanimously agreed that it is mustahabb to do ghusl for Eid prayer.
The reason why it is mustahabb is the same reason as that for doing ghusl before Jumuah and other public gatherings. Rather on Eid the reason is even stronger.
2 – Eating after the prayer on Eid al-Adha:
But on Eid al-Adha it is mustahabb not to eat anything until one comes back from the prayer, so he should eat from the udhiyah if he has offered a sacrifice. If he is not going to offer a sacrifice there is nothing wrong with eating before the prayer.
3 – Takbeer on the day of Eid
This is one of the greatest Sunnahs on the day of Eid because Allaah says (interpretation of the meaning):
"(He wants that you) must complete the same number (of days), and that you must magnify Allaah [i.e. to say Takbeer (Allaahu Akbar: Allaah is the Most Great)] for having guided you so that you may be grateful to Him"
[al-Baqarah 2:185]
It was narrated that al-Waleed ibn Muslim said: I asked al-Awzaai and Maalik ibn Anas about saying Takbeer out loud on the two Eids. They said, Yes, Abd-Allaah ibn Umar used to say it out loud on the day of al-Fitr until the imam came out (to lead the prayers).
Al-Daaraqutni and others narrated that on the morning of Eid al-Fitr and Eid al-Adha, Ibn Umar would strive hard in reciting takbeer until he came to the prayer place, then he would recite takbeer until the imam came out.
Ibn Abi Shaybah narrated with a saheeh isnaad that al-Zuhri said: The people used to recite Takbeer on Eid when they came out of their houses until they came to the prayer place, and until the imam came out. When the imam came out they fell silent, and when he said takbeer they said takbeer. See Irwa' al-Ghaleel, 1/121
Saying takbeer when coming out of ones house to the prayer place and until the imam came out was something that was well known among the salaf (early generations). This has been narrated by a number of scholars such as Ibn Abi Shaybah, 'Abd a l-Razzaaq and al-Firyaabi in Ahkaam al-Eidayn from a group of the salaf. For example, Naafi ibn Jubayr used to recite takbeer and was astonished that the people did not do so, and he said, "Why do you not recite takbeer?"
Ibn Shihaab al-Zuhri (may Allaah have mercy on him) used to say, "The people used to recite takbeer from the time they came out of their houses until the imam came in."
The time for takbeer on Eid al-Fitr starts from the night before Eid until the imam enters to lead the Eid prayer.
In the case of Eid al-Adha, the takbeer begins on the first day of Dhul-Hijjah and lasts until sunset on the last of the days of tashreeq.
Description of the takbeer:
It was narrated in the Musannaf of Ibn Abi Shaybah with a saheeh isnaad from Ibn Masood (may Allaah be pleased with him) that he used to recite takbeer during the days of tashreeq:
Allaahu akbar, Allaahu akbar,
laa ilaaha ill-Allaah,
wa Allaahu akbar,
Allaah akbar, wa Lillaah il-hamd
(Allaah is Most Great, Allaah is most Great, there is no god but Allaah, Allaah is Most great, Allaah is most great, and to Allaah be praise).
It was also narrated elsewhere by Ibn Abi Shaybah with the same isnaad, but with the phrase "Allaahu akbar" repeated three times.
Al-Mahaamili narrated with a saheeh isnaad also from Ibn Masood: "Allaahu akbaru kabeera, Allaahu akbaru kabeera, Allaahu akbar wa ajallu, Allaahu akbar wa Lillaah il-hamd
(Allaah is Most Great indeed, Allaah is Most Great indeed, Allaah is most Great and Glorified, Allaah is Most Great and to Allaah be praise)." See al-Irwa', 3/126.
4 – Offering congratulations
The etiquette of Eid also includes the congratulations and good wishes exchanged by people, no matter what the wording, such as saying to one another
Taqabbala Allaah minna wa minkum
(May Allaah accept (good deeds) from us and from you" or "Eid mubaarak" and other permissible expressions of congratulations.
It was narrated that Jubayr ibn Nufayr said: When the companions of the Prophet (peace and blessings of Allaah be upon him) met one another on the day of Eid, they would say to one another, "May Allaah accept (good deeds) from us and from you." Ibn Hajar said, its isnaad is hasan. Al-Fath, 2/446.
Offering congratulations was something that was well known among the Sahaabah, and scholars such as Imam Ahmad and others allowed it. There is evidence which suggests that it is prescribed to offer congratulations and good wishes on special occasions, and that the Sahaabah congratulated one another when good things happened, such as when Allaah accepted the repentance of a man, they went and congratulated him for that, and so on.
Undoubtedly these congratulations are among the noble characteristics among the Muslims.
5 – Adorning oneself on the occasion of Eid.
It was narrated that Abd-Allaah ibn Umar (may Allaah be pleased with him) said that Umar took a brocade cloak that was for sale in the market and brought it to the Messenger of Allaah (peace and blessings of Allaah be upon him), and said, "O Messenger of Allaah, buy this and adorn yourself with it for Eid and for receiving the delegations." The Messenger of Allaah (peace and blessings of Allaah be upon him) said to him, "Rather this is the dress of one who has no share (of piety or of reward in the Hereafter)…" Narrated by al-Bukhaari, 948.
The Prophet (peace and blessings of Allaah be upon him) agreed with Umar on the idea of adorning oneself for Eid, but he denounced him for choosing this cloak because it was made of silk.
It was narrated that Jaabir (may Allaah be pleased with him) said: The Prophet (peace and blessings of Allaah be upon him) had a cloak which he would wear on the two Eids and on Fridays. Saheeh Ibn Khuzaymah, 1756,
Al-Bayhaqi narrated with a saheeh isnaad that Ibn Umar used to wear his best clothes on Eid.
So a man should wear the best clothes that he has when going out for Eid.
With regard to women, they should avoid adorning themselves when they go out for Eid, because they are forbidden to show off their adornments to non-mahram men. It is also haraam for a woman who wants to go out to put on perfume or to expose men to temptation, because they are only going out for the purpose of worship.
6 – Going to the prayer by one route and returning by another.
It was narrated that Jaabir ibn Abd-Allaah (may Allaah be pleased with him) said: On the day of Eid, the Prophet (peace and blessings of Allaah be upon him) used to vary his route. Narrated by al-Bukhaari, 986.
It was said that the reason for that was so that the two routes would testify for him on the Day of Resurrection, for the earth will speak on the Day of Resurrection and say what was done on it, both good and bad.