Do you pray all sunnah silently? and if it is, do you have to move your lips while reciting a surah or just think when you're reciting? (sorry for my grammar, i hope you will still understand this)
The sunnah prayers similar to all prayers must be performed in the manner the prophet () showed us.
This means almost all sunnah and nafl prayers we usually perform in a daily manner like the sunan ar-Rawatib (see for example What are Sunnah Rawaatib?), fajr, shafa'a, witr, doha etc. should be performed silently as almost all of them should be performed alone.
On the other hand any prayer which can be performed in congregation such as 'Id prayers, Taraweeeh or Tahajjud and many more should be performed "loudly" or more exactly in audible voice. This may include the shafa' and witr in Ramadan.
As a matter fo fact reciting the qur'an without moving the lips be it in silent prayer or "loud" prayer (and while reading it) is regarded by the majority of scholars as an order and not doing so may invalidate the prayer.