Well, Paul did go on the sabbath to the synagogue to preach to Jews.
That is because Paul wanted to see Israel saved.
Romans 10:1
Brethren, my heart's desire and prayer to God for Israel is, that they might be saved.
If they will be found in the synagogue studying, Paul is just going where he will find the people he is looking for. Who better to start with than the people who already know Scripture. The churches that were started on Paul’s missionary journeys weren’t brand new out of Gentile box churches. Many of the believers were followers of Scripture already When they were introduced to the truth of Jesus as the Christ, some people rejected and some believed. There were Gentiles saved but Paul went first to the synagogues wherever they were and afterwards to the Gentiles.
Acts 18
Paul was pressed in the spirit, and testified to the Jews that Jesus was Christ. And when they opposed themselves, and blasphemed, he shook his raiment, and said unto them, Your blood be upon your own heads; I am clean: from henceforth I will go unto the Gentiles.
It is also seen as his travels are recorded that as he went, he looked for the synagogues.