All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

The Prophet said, “Whoever fasts the month of Ramadan and then follows it with six days of Shawwaal will be [rewarded] as if he had fasted for a whole year.” [Muslim]

In another narration the Prophet said: “Whoever fasts six days after ‘Eed-Ul-Fitr will be rewarded as if he had fasted the entire year. Whoever does a good act will be rewarded ten times.” [Ibn Maajah]

Fasting the missed days of Ramadan first is preferable as the reward in the aforementioned narrations is for the one who fasts the entire month of Ramadan and then fasts six days of Shawwaal. Therefore, you should make up the missed days of Ramadan first and then fast the six days of Shawwaal.

This is a controversial issue, but the most preponderant and most sound opinion is that it is lawful to fast the six days of Shawwaal while delaying the missed days of Ramadan.

The above rule concerns only the missed days of Ramadan that were not fasted for some legitimate excuse.  However, if the person did not fast some days in Ramadan without having an Islamicaly legal excuse, then he/she should make up those days before any supererogatory fasting. This is so because the person who does so has transgressed the limits of Allaah.

Allaah Knows best.