If I intend to fast 6 days after Ramadhan, do I need to fast on the day of Eid also or after Eid?

Fasting on Eid day is haraam. [1] The Prophet (ﷺ) forbade us to fast on the days of Eid and on Fridays by themselves. [2]

References:

[1] The Prophet (ﷺ) forbade fasting on the day of al-Fitr and al-Nahr.” Narrated by al-Bukhari, 1992; Muslim, 827. The scholars are unanimously agreed that fasting on these two days is haraam. 

[2] Abu Hurayrah (radiAllahu anhu) that he said: “I heard the Prophet (ﷺ) say, ‘None of you should fast on a Friday unless he fasts the day before or the day after.’” (Narrated by al-Bukhari, 1849, Muslim, 1929)