Doing ghusl before going for Eid prayer: It is mustahabb to do ghusl before going for Eid prayers. Just like it is recommended for Jumu’ah, a Muslim is advised to observe it. It was narrated in a saheeh hadeeth in al-Muwatta’ and elsewhere that ‘Abd-Allah 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.
Also, It is a day we must emphasis that fasting has ended, so, endeavour to eat something. It is strongly recommended even it it just little, you need to take in food, snacks or fruits. The Prophet (s.a.w) used not to go out on the morning of Eid al-Fitr until he had eaten some dates… of which he would eat an odd number. (Al-Bukhaari) This however does not apply to Eid-il-Adha, which does not permit eating before going to Eid until one comes back from the prayer.
Do Takbeer on the day of Eid: This is one of the greatest Sunnahs on the day of Eid because Allah says (interpretation of the meaning):”(He wants that you) must complete the same number (of days), and that you must magnify Allah [i.e. to say Takbeer (Allahu Akbar: Allah is the Most Great)] for having guided you so that you may be grateful to Him” [al-Baqarah 2:185]
Allahu akbar, Allahu akbar, laa ilaaha ill-Allah, wa Allahu akbar, Allah akbar, wa Lillaah il-hamd (Allah is Most Great, Allah is most Great, there is no god but Allah, Allah is Most great, Allah is most great, and to Allah be praise). It was narrated that al-Waleed ibn Muslim said: I asked al-Awzaa’i and Maalik ibn Anas about saying Takbeer out loud on the two Eids. They said, Yes, ‘Abd-Allah ibn ‘Umar used to say it out loud on the day of al-Fitr until the imam came out (to lead the prayers).
Offering congratulations/Salutation. The etiquette of Eid also include good wishes by people, no matter what the wording, such as saying to one another Taqabbala Allah minna wa minkum (May Allah accept (good deeds) from us and from you” or “Eid mubaarak” and other permissible expressions of congratulations.
Adorning oneself on the occasion of Eid: Every Muslim is advised to wear his or her best clothes. But women must not show off their beauty to commit sin.
Lastly, going to the prayer by one route and returning by another important etiquette. On the day of Eid, the Prophet (peace and blessings of Allah be upon him) used to vary his route. (al-Bukhaari)
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.