1. Acts 9 gives us Paul's "conversion" to Christ. It is clearly a turning point in his life, a radical change of direction. He goes from persecuting believers (cf. Phil. 3:6) to being one. He does not stop being a Jew. He simply believes Jesus is the promise Messiah of Israel. He does not start going by Paul, as if that's his Christian name. No that shift doesn't happen for four more chapters and maybe 15 years.
Paul is likely his Roman name or nickname. If I had to pick a psychological scenario, it's that as a Diaspora Jew, he had been running from his Hellenistic identity. He didn't want to be a second rate Israelite. He was a "Hebrew of Hebrews" (Phil. 3:5; 2 Cor. 11:22). He had been distancing himself from being born way off in a place like Tarsus his whole life, a Greek-speaker. He had been distancing himself from his Roman citizenship. He becomes a Pharisee.
This is a different narrative than the one William James (1842-1910) suggested, reading Paul as an older version of Luther. In that version, Paul struggles so much with a guilty conscience that he finally can't take it any more and comes to the conclusion that God accepts us only on the basis of his grace, his unmerited favor on us. The problem is that Paul thought himself blameless in his law-keeping before he believed on Jesus (Phil. 3:6).
But Jesus comes and taps Jesus on the shoulder. "You know you think you've been helping God out," Jesus in effect tells him. "Well, you're actually fighting against him."
2. A man named Ananias at a city called Damascus lays hands on Paul. He receives the Holy Spirit. He is baptized. He instantly changes from persecutor to believer. He had ironically gone to that far north city to arrest some believers.
Damascus is way north, about as far north of the Sea of Galilee as Jerusalem is south, in modern day Syria. It was way out of Jerusalem's "jurisdiction," so to speak. Must have been a little on the sly? It probably wasn't a "go arrest any Christians you can find" but more "Go get so and so whose in big trouble."
Acts does not tell us what Paul tells us in Galatians 1. Paul lets three years go by before he returns to Jerusalem (Gal. 1:17-18). That makes sense. His former "employers" wouldn't be too happy with him. Instead, he goes east just outside the Roman empire to Arabia, probably the city of Petra, just east of Damascus.
Since individuals from that Arab kingdom come after Paul three years later (2 Corinthians 11:32-33), he surely ended up doing some preaching there that ticked someone off. It's easy to picture Paul preaching something like Jonah in Nineveh. Repent or God's going to fry you all. It probably wouldn't have made anyone too happy.
3. We get a small sense of Acts' point of view in the way it tells of Paul's escape from Damascus. Acts says it is the Jews who are trying to kill Paul (9:23). This is not Paul's perspective. Paul says that it is the ethnic leader of the Arabs in Damascus who was charged to arrest him and presumably take him back to Petra (2 Cor. 11:32). This gives us a small hint of how Acts tends to blame the Jews for Paul's troubles and downplay the trouble he ran into with secular authorities like the Romans.
The book of Acts is in some sense an "apology," a defense of the earliest believers. It has as a key perspective that the Early Christians were not troublemakers and tends to pin the blame on troubles with local authorities on opposition within the Jewish communities. This brief incidence gives us a significant insight into this perspective.