Praise be to Allah.
You can start fasting six days of Shawwaal from the second day of Shawwaal, because it is haraam to fast on the day of Eid. You can fast the six days at any time during Shawwaal, although the best of good deeds are those which are done soonest.
The standing committee received the following question:
Should fasting the six days be done immediately after Ramadaan, following the day of Eid or is it permissible to do it a few days after Eid in the month of Shawwaal or not?
They replied as follows:
These days do not have to be fasted immediately after Eid al-Fitr; it is permissible to start fasting them one or more days after Eid, and they may be done consecutively or separately during the month of Shawwaal, according to what is easier for a person. There is plenty of room for maneuver in this matter, and this is not obligatory, it is Sunnah.
And Allaah is the Source of strength. May Allaah bless our Prophet Muhammad and his family and companions and grant them peace.