I want to know if I could start fasting the 6 days of Shawwal before or after making up the days I didn't fast in Ramadan?

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

The best thing to do for a person who has to make up for missed days of Ramadan, is to make up for the missed days of Ramadan before starting supererogatory fast (fasting the days of Shawwaal), since it is desirable to perform one's obligation as early as possible. Imam Ahmad  said: “It is not permissible for one to fast non-obligatory fast before making up the missed obligatory fast”.

Allah knows best.