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.

'Some of the students of al-Buhlul said: A letter was handed to al-Buhlul so he opened it and it was in it:

"From a woman from Samarqand of Khurasan [near Afghanistan], I am a woman who committed all types of sins that no other ever committed, but turned to Allah and repented, and I asked about the worshippers living on earth so I was informed of four worshippers, one of them is al-Buhlul in Africa. O Buhlul, I ask you by Allah to invoke Allah for me so He maintain for me this guidance"

1. Eat, drink and be moderate
Almost all of us do it - once Iftar time hits, we just keep plowing food and drink into our mouths till it's hard to move afterwards. And those of us who do it know this is totally contrary to the spirit of Ramadan, through which we're supposed to learn self-control not self-indulgence. Let's try to stick to the Prophetic rule on eating: fill our stomachs with one-third food, one-third water and one-third breathing space, even in Ramadan.