What does Acts 9:20 mean?

The timeline here is a bit confusing, but it appears that Saul is run out of Damascus by the Jewish leaders, then goes to Arabia (Galatians 1:17), which is quite close to Damascus. He is there for "many days," detailed as three years in Galatians 1:18. He returns to Damascus and the governor under King Aretas apparently threatens him and he escapes through the city wall in a basket again (2 Corinthians 11:32–33). It is possible the threat from the Jewish leaders and the governor is the same. Either way, Saul escapes, and three years after his conversion (Galatians 1:18), makes his way to Jerusalem (Acts 9:26; Galatians 1:18). Luke is often more specific in his timelines, but only when he has exact information to work with or when he is with Saul personally.

Synagogues were established around the time of the Babylonian captivity. The temple in Jerusalem had been destroyed, and the Jews were scattered all over the Babylonian Empire. The term synagogue is from the Greek for "bringing together," and this is the place people gathered to read from the Hebrew Scriptures—our Old Testament—and discuss what they read.

At this time, most of the Christians are Jews. They believe that Jesus is the Jewish Messiah and worshiping Him is the natural fulfillment of their childhood religion. It makes sense, then, that Saul first goes to the local area's main meeting place for prayer. That place would be a synagogue, unless there are not enough Jews in the city. Saul, who later goes by his Greek name, Paul, does this in every town he visits (Acts 14:1; 16:13; 17:1–2, 10; 18:4; 19:8). The people are already familiar with the Hebrew Scriptures that prepared the way for the coming of the Messiah. Saul's job is to connect the dots.