Paul's own account of his change is short and strange. In Galatians he says he persecuted the church violently, that God revealed his Son to him, that he did not consult anyone, went into Arabia, and only after three years went up to Jerusalem, where he saw Peter and James and no other apostle. He is at pains to insist he received his commission from no human being, because his standing was under attack.
Acts tells it three times, with growing detail: the light on the Damascus road, the voice, the blindness, Ananias. The three tellings within Acts differ from each other on whether Paul's companions heard the voice or saw the light, and all three differ from Paul's own compressed version in emphasis and sequence. The word conversion fits awkwardly in any case: Paul does not describe leaving one religion for another. He describes being called, in the language the Hebrew prophets use of themselves, and he continues to identify as an Israelite of the tribe of Benjamin.
The argument he never stopped having was about Gentiles. Did a non-Jew joining the movement have to be circumcised and keep the law? Paul said no, and said it with a ferocity that survives translation, in Galatians he wishes those pushing circumcision would go the whole way and castrate themselves. He records confronting Peter to his face at Antioch over table fellowship. Acts smooths this into a council that agrees amicably.
That Paul won this argument is why Christianity became a religion of the empire rather than a Jewish sect. It is also why the New Testament exists in Greek.