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.
Do the six days of Shawwaal have to be fasted consecutively?
Praise be to Allaah.
It is not a necessary condition that they should be fasted consecutively. If you fast them separately or consecutively, it is OK. The sooner you do them, the better, because Allaah says (interpretation of the meanings):
so compete in good deeds [al-Maaidah 5:48]
And march forth in the way (which leads to) forgiveness from your Lord [Aal Imran 3:133]
[Moosa peace be upon him said:] and I hastened to You, O my Lord, that You might be pleased [Ta-Ha 20:84]
And (you should hasten to fast these six days) because delaying may cause problems. This is the view of the Shaafa is and some of the Hanbalis, but it is OK if you do not hasten it and you delay it until the middle or end of the month.
Al-Nawawi (may Allaah have mercy on him) said:
Our companions said: it is mustahabb to fast six days of Shawwaal. Because of this hadeeth they said: it is mustahabb to fast these days consecutively at the beginning of Shawwaal, but if one separates them or delays them until after Shawwaal, this is permissible, because he will still be following the general guidelines of the hadeeth. We have no dispute regarding this matter, and this was also the view of Ahmad and Dawood.
Al-Majmoo Sharh al-Muhadhdhab