The next few chapters may be confusing as Saul is referred also as Paul-see verse 19 here, but also chapter 11 v 25. The book of Acts is Luke's account of the spread of the Early Church and here of Paul's conversion and the first years after that, so it is good to compare this to Paul's own account in Galatians 1 v 15-24 for an all-round view. When putting together the two records we can note this chronological order of events from the time when Saul sets off on the road to Damascus:
1. He is converted on the road.
2. He preaches in Damascus.
3. He goes away to Arabia
4. He returns and preaches in Damascus for three years.
5. He goes to Jerusalem
6. He escapes from Jerusalem to Caesarea
7. He returns to the regions of Syria and Cilicia.
We read here that Paul immediately witnessed to his faith in Damascus, where there were many Jews-and so Paul headed for the Synagogues. The man returns from where he frequented as a Pharisee, but he is a changed man.
Secondly, he had to spend time alone with God. Just like Jesus his Master he needed two things before he began this new life: guidance and strength, supernatural strength.
The Jews recognised what had happened to Paul as they dispatched people to murder him. This continued for THREE years, the Jews thinking they had Paul in a trap, but he makes his escape over the walls of that ancient city. Paul showed great courage in standing up to those who had killed Stephen and the fact that they tried to kill him demonstrates the efficacy of his preaching. No one attacks the person who is ineffective. Verse 31 speaks to the reader of God's protection.