When could you start to pray? Is it at the start of the Azan or after it?

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

When the prayer time begins, a Muslim is permitted to perform the prayer, and the Athaan is a sign that the prayer time started.  

Therefore, a Muslim may perform the prayer as soon as he hears the Athaan but it is preferable to repeat it as the Prophet ordered us to do so.  Abdullaah Ibn Umar narrated that he heard the Prophet saying: ''When you hear the Muaththin (person calling Athaan), you should repeat what he says, and then ask Allaah to exalt my mention, because whoever exalts my mention one time, Allaah will exalt his mention ten times, then ask Al-Waseelah (a very high rank in Paradise) for me, as only one of Allaah's Slaves deserves it and I hope that it will be me. Whoever asks Al-Waseelah for me, he will find my intercession.'' [ Muslim].      

Allaah knows best.