As part of a conversation elsewhere:
In this particular case, Saul was going to the synagogues in Damascus to arrest any Jews who "belonged to the Way", in the name of the High Priest (Acts 9:1-2).
You may recall, at this time, the name "Christian" did not yet exist, and the only Christians in existence were Jews (who observed the Torah and who had not quite gotten beyond their belief that the Messiah was only for Jews and for converts to Judaism (involving circumcision for the males and immersion for both sexes).
By Acts 15, a loud segment of the Way was composed of Pharisees who insisted on this conversion process for non-Jews (Acts 15:5).
When the Good News reached Phillipi, Paul and his companions did not find a synagogue (probably because there weren't enough Jewish men in town to fulfill the minimum number required to have a synagogue, because they likely had been expelled from this Roman colony (Acts 16:12) when Emperor Claudius expelled all the Jews from Rome (Acts 18:2)). So they went to the river-side, where by tradition would be a back-up location for Jewish assemblies. There they found women, one of whom was Lydia, who after conversion, opened her house up as an assembly place (Acts 16:15,40).
In Thessalonica, the Jews who were jealous of Paul's, et al, popularity, expected to find him in Jason's house (Acts 17:1-9). They did not find Paul and his crew there, but they did find "some of the brothers" (v 6). Perhaps these brothers were some of those who had heard Paul & Silas teaching in the synagogue, who "were persuaded and joined Paul and Silas", along with God-fearing non-Jews and women (v 4).
In Corinth, Paul spent all his time reasoning (note, not "lecturing") in the synagogue (Acts 18:4) with both Jews and Greeks. When his listeners resisted, Paul left the synagogue and started meeting next door in Titius Justus' house (vv 6-7). Crispus, the leader of the synagogue, apparently went with them (v 8), and his successor, Sosthenes (who was a co-writer of 1 Cor (1 Cor 1:1)), seems to have become a believer but remained at his post as new leader of the synagogue (v 17) (or alternatively, the Christian assembly was referred to as "synagogue" (as also found in James 2:2) and had a "leader").
In Ephesus, Paul engaged in discussion (note, not "lectured") with the Jews in the synagogue (Acts 18:19), who wanted him to stay around a while. He had to leave, but Priscilla and Aquila still hung out in the synagogue, which is where they first heard the Jew Apollos teaching about Jesus (Acts 18:24-26). Later Paul returned to Ephesus, where he engaged in discussion (note, not "lectured") in the synagogue for 3 months (Acts 19:1-8), until the Jews became hardened, at which time Paul broke away and "began a new congregation" in the lecture hall of Tyrannus (v 9).
And so forth and so on.
So at least in the early days, if you wanted to find an assembly of Christ, you started at the local Jewish meeting place, and went from there.