There is not just one particular part of the prayer when du’a should be offered, rather there are several parts of the prayer where Prophet Muhammad offered du’a from the opening takbir to the final taslim.

It is proven that the Prophet (peace and blessings of Allah be upon him) used to point with his index finger and move it during the tashahhud when praying.

If you are reciting the Surah after Al-Fatihah from the beginning, then you should say the Basmalah (Bismillah ir-Rahman ir-Rahim) apart from Surah At-Tawbah. If you are reciting from the middle of a Surah, then it is not recommended to recite the Basmalah.