You can start fasting six days of Shawwal from the third day of Shawwal, because it is haram to FAST on the day of 'Eid. You can FAST the six days at any time during Shawwal, although the best of good deeds are those which are done soonest. 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)
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
For details: http://sunnahonline.com/library/hajj-umrah-and-the-islamic-calendar/322-fasting-the-six-days-of-shawwal