Jazakhalah Khayran bother for seeking knowledge about the sunnah of Rasul Allah (P.B.U.H).
Muhammed (Pb.u.h) would fast another 6 days of shawaal... after ramadan. Baiscally.. what yooh can do is fast, for instance 3 weeks of saturdays and sundays.. and then you'll get your 6 days of shawaal fasting.. or on the other hand.. you could fast six straight days,.. which i would recommend easier..
The rewards for fasting the 6 days of shawaal is the same reward of you FASTING A WHOLE YEAR.. subhanalah... imagine all this hassanah ya akiee...
The Second Type Of Fast Rasoul Allah (p.b.u.h) would fast is: the first Mondays and Thrusday's of every month...
As in a hadith it is mentioned ... ( sorry cant find that hadith right now..)