My Bible states that it was Paul's custom to go to "the Jew first". (Romans 1:16) Paul's burden after conversion was to preach Christ. He was burdened for his own people, the Jews. So he went where the Jews were to preach Christ to them.
Acts 13:5 And when they were at Salamis, they preached the word of God in the synagogues of the Jews: and they had also John to their minister.
Acts 13:14 But when they departed from Perga, they came to Antioch in Pisidia, and went into the synagogue on the sabbath day, and sat down.
Acts 13:15 And after the reading of the law and the prophets the rulers of the synagogue sent unto them, saying, Ye men and brethren, if ye have any word of exhortation for the people, say on. (Notice, Paul did not do the "reading of the law and the prophets")
Acts 13:16 Then Paul stood up, and beckoning with his hand said, Men of Israel, and ye that fear God, give audience.
Acts 13:42 And when the Jews were gone out of the synagogue, the Gentiles besought that these words might be preached to them the next sabbath.
Acts 13:43 Now when the congregation was broken up, many of the Jews and religious proselytes followed Paul and Barnabas: who, speaking to them, persuaded them to continue in the grace of God.
Acts 13:44 And the next sabbath day came almost the whole city together to hear the word of God.
Acts 16:13 And on the sabbath we went out of the city by a river side, where prayer was wont to be made; and we sat down, and spake unto the women which resorted thither.
Acts 16:14 And a certain woman named Lydia, a seller of purple, of the city of Thyatira, which worshipped God, heard us: whose heart the Lord opened, that she attended unto the things which were spoken of Paul. (Paul preached the Gospel, not the law)
Acts 17:15 Now when they had passed through Amphipolis and Apollonia, they came to Thessalonica, where was a synagogue of the Jews:
Acts 17:2 And Paul, as his manner was, went in unto them, and three sabbath days reasoned with them out of the scriptures,
Acts 17:3 Opening and alleging, that Christ must needs have suffered, and risen again from the dead; and that this Jesus, whom I preach unto you, is Christ.
Acts 17:4 And some of them believed, and consorted with Paul and Silas; and of the devout Greeks a great multitude, and of the chief women not a few.
Acts 18:4 And he reasoned in the synagogue every sabbath, and persuaded the Jews and the Greeks.
Acts 18:5 And when Silas and Timotheus were come from Macedonia, Paul was pressed in the spirit, and testified to the Jews that Jesus was Christ.