There's no correct way to recite the qur'an.
For qualifying a correct manner for the recitation you need to be more specific:
Are you male or female is there an audience or not? If so what kind of audience (gender, belief etc.)? What is your intention when you recite? Are you listening to your recitation or trying to apply the rules of tajweed or just reading and maybe looking in a Tafseer or translation to understand what you read/recite...
In most of the above cases it is better to recite in a louder voice so that you may hear and listen to your own words. While in some cases it might not be good or even disliked. But in general it is recommended (some scholars may consider it as rather essential) to move the lips while reciting/reading the qur'an.