Allah (`azza wa jall) says in the Qur’an, "Say (O Muhammad): ‘If you (really) love Allah then follow me, Allaah will love you and forgive you your sins. And Allah is Oft-Forgiving, Most Merciful."
[surah ali-Imran, 3: 31]
This is a beautiful verse, named by some of the salaf as "the verse of the test", as it tests how true one’s love of Allaah is. They explained that if one loves Allah, then he must show that in his/her following of the Prophet Muhammad (salAllahu alayhi wasalam). The verse tells us that those who follow the Prophet (salAllahu alayhi wasalam) if sincere, can insha'Allah expect the following two:
One of the ways to manifest our loving of Allah, by following the Prophet (salAllahu alayhi wasalam) is to do those acts that he (salAllahu alayhi wasalam) advised his Companions, and the Ummah in general, to do. A sunnah which is certainly relevant to us in these days is his(salAllahu alayhi wasalam) practice to fast six days in the month of Shawwal.
Abu Ayyub al-Ansari narrated that Allah’s Messenger (salAllahu alayhi wasalam) said: "He who fasts Ramadan, and six of Shawwal, it will be (in terms of rewards) as if the fasted a whole year."
[Muslim, at-Tirmidhi, Abu Dawud, Ahmad, Ibn Majah]
So this is an established sunnah, which carries a great reward.
In commenting on the above mentioned hadith, As-San'ani said in Subul us-Salam: "If the thirty days of Ramadan fasting are assimilated with the six days of fasting in Shawwal, it altogether makes 36 days. According to Shari`ah, each virtue is rewarded ten times. Therefore, if we multiply 36 with 10, it makes 360, a number which equals the days of a year. Some scholars are of the opinion that these six days of fasting in Shawwal must be completed in a continuous order right after the end of Ramadan. Others believe that is enough to merely complete six days of fasting in Shawwal (in any order, either successive or with intervals), an opinion which is deemed to be correct." We may also fast on Mondays and Thursdays, as in that case we would be following another Sunnah: A’isha (radiAllahu anha) narrated: "The Messenger of Allah (salAllahu alayhi wasalam) used to fast Mondays and Thursdays".
If it is easier for one to fast on weekends, then in that case one would still be following another sunnah at the same time: Umm Salama (radiAllahu anha)narrated that Allah’s Messenger used to fast mostly on Saturday and Sunday, and he used to say: "They are the festival days for the mushrikeen, and I like to act contrary to them."
[an-Nasa’i, Ibn Khuzaymah, who graded it saheeh, and Ibn Hajar agreed]