Abu Ayyub al-Ansari (may Allah be pleased with him) reported Allah’s Messenger (SAW) as saying: He who observed the fast of Ramadan and then followed it with six (fasts) of Shawwal, it would be as if he fasted perpetually. – Sahih Muslim 1164
It was the sunnah of the Prophet to fast it and to love Allah is to follow the example of the Prophet (SAW).
Narrated Abu Hurairah: Allah’s Messenger (SAW) said, (Allah says about the fasting person), ‘He has left his food, drink and desires for My sake. The fast is for Me. So I will reward (the fasting person) for it and the reward of good deeds is multiplied (at least) ten times.” – Sahih Bukhari 1894
So if the fasts in Ramadan are rewarded multiplied by ten (at least), that will be 290 or 300 good deeds. Adding the six days in Shawwal makes it 350-360 good deeds which are roughly equivalent to the number of days in a year.
It is acceptable to fast the next 6days after Eid, or to spread them out throughout the month in no particular order with care not to single out Fridays. They could also be made to coincide with the white days (13th, 14th and 15th) of the month or Mondays and Thursdays.
If we have some days we missed in Ramadan, we can make them up before we commence the 6 of Shawwal.
May Allah guide us to what is good and make it easy to comply.