Ayyub (Allah be pleased with him) relates that the Messenger of Allah (blessings and peace be upon him, his family, and companions) said, “Whoever fasts Ramadan and follows it with six days from Shawwal it is as if they fasted the entire year.” [Muslim, Abu Dawud, Tirmidhi, and Ibn Majah; Ahmad transmitted it from Jabir, Muntaqa]
Based on this, the majority of the fuqaha, including the Hanafi, Shafi`i, and Hanbali school, have held that it is recommended to fast six days of the month of Shawwal. [Radd al-Muhtar; Mughni al-Muhtaj Shah al-Minhaj; Kashshaf al-Qina`]
It is prohibitively disliked (makruh tahriman) and sinful to fast on Eid day itself. [Radd al-Muhtar]
that should cover it !