After Paul left Athens he came to Corinth, and as was his manner, he began to share Christ with his Jewish brethren in the local synagogue (Acts 18:4). This may also have been how he actually met Aquila and Priscilla, because Jesus told his disciples that when they entered a city to first inquire who in that city was hospitable enough to house guests and stay there (cp. Matthew 10:11). What better place could there be for a Jew to find hospitable Jews than the local synagogue? Archeology has uncovered in Corinth a partial inscription in Greek on a lintel, which is believed to have read (when complete) Synagogue of the Hebrews. Its writing indicates a later structure, but the synagogue over whose doorway this lintel was placed may have stood upon the same foundation of that in which Paul preached.
It was there that, Sabbath after Sabbath, Paul discussed and debated with the folks—both Jews and God-fearing Greeks that Jesus is the Christ. Codex D (the Western text) expands upon verse-4 by saying that Paul inserted Jesus’ name in certain prophecies as “an interpretive expansion in those passages.” Even if these insertions into the Western text are not part of the original, this shows exactly how Jews of Paul’s day read the Scriptures in an interpretive manner. The Targum Jonathan, which is a Jewish interpretive Greek translation of the Prophets, often inserts the Word of the Lord in the text where the Tetragrammaton (YHWH) appears in the Hebrew. Even today, all Jews use the word Lord wherever the Tetragrammaton appears in Hebrew Scriptures. The Targum Jonathan also inserts the Greek word for Messiah after “my servant” in Isaiah 42:1 and 52:13. In fact, Paul uses this method of interpretation in his epistle to the Corinthians when he writes of the Rock which Moses struck to give Israel water (cp.1Corinthians 10:4). So, this Jewish manner of interpreting the text does indicate the manner in which Paul probably shared or interpreted Christ in the Scriptures while ministering at Corinth.
We don’t know how long Paul was permitted to preach Christ in the synagogue, but it was probably longer than he was able to do so at Thessalonica, because, Crispus, the ruler (or one of the rulers (cp. Acts 18:8, 17) of the synagogue at Corinth believed. When the leaders of a community accept something as true, more consideration is given to the matter by the rest of the community. This is why the leaders are so important, and why they are held so responsible by God, if the people are led astray.
Luke also mentions a God-fearer named Titius Justus (Acts 18:7), who is probably the same person Paul calls Gaius in 1Corinthians 1:4, who was one of the few Paul baptized. This would make his full name Gaius Titius Justice and a Roman citizen (Corinth was Roman colony). Paul referred to him in his letter to the Romans as his host (i.e. where Paul dwelled while in Corinth at the time of his writing the epistle), and he was also the host of the church at Corinth. Luke tells us his house joined hard to the synagogue there (i.e. both Gaius’ house and the synagogue probably shared the same wall, cp. Acts 18:7).
Luke goes on to say that when Paul left the synagogue he was given use of Gaius’ home. That is, no only had it become the place where the church met, but Paul used it as the center of his outreach, similar to how he used the School of Tyrannus in Acts 19:9. Luke does not mean to say, as some have thought, that Paul left the home of Aquila and Priscilla at this time as well in order to live in Gaius’ home. Paul continued to dwell with Aquila and Priscilla during his stay at Corinth, but worked out of the home of Gaius. Only later, near the end of his third recorded missionary journey, when Aquila and Priscilla were dwelling in Rome (cp. Romans 16:3) did Paul dwell in the house of Gaius (cp. Romans 16:23).