We know Paul was against Gentile circumcision as was the church council but he was willing to have Timothy be circumcised for the sake of the ministry, not as a belief that anyone would have to be circumcised to be saved.
Paul, Silas, and Timothy then went from town to town sharing the instruction handed down from the church in Jerusalem. Doing that gave the other churches correct information and encouragement and the churches were strengthened and grew in number.
They continued on through Phrygia and Galatia because the Holy Spirit prevented them from going into Asia. They came to the border of Mysia, headed north toward Bithynia. (All of these areas were districts of what is now Turkey.) However, the Holy Spirit would not let them go into Bithynia and instead they went to the seaport of Troas. That very night, Paul had a vision of a man from Macedonia in Greece who pleaded with him, "Come over to Macedonia and help us!"
Notice in 16:1-9, the writing uses the terms they, he, Paul, Silas, Timothy, etc. Beginning with verse 10, you will notice the writer uses we and us. Knowing Luke is the author of Acts, we realize something has changed. During the writing about Paul and Timothy coming to Troas, Luke joined them in Troas and went with them on this journey.
Paul, Silas, Timothy and Luke went on to Philippi which was a city in Macedonia and a Roman colony. They stayed there several days but on the Sabbath they went outside the city to a riverbank where people might gather for prayer. They met a woman named Lydia who was a purple cloth merchant from Thyatira and a worshipper of God which was another way to say Gentile. As she listened to Paul, the Lord opened her heart and she accepted his teaching about Jesus the Messiah. She believed and was baptized. She told them if they truly believed she had become a believer in Christ, they would have to come and stay at her home, so they went with her. Her entire household, including servants, also became believers in Christ.
(I wondered why she may have insisted they stay with her if they "truly" believed she had become a believer. One reason might be gratitude, but I also wonder if it had something to do with what Jesus experienced with the Jews. When Jesus ate with people who were Gentiles or tax collectors, etc., He was criticized heavily by the Jewish religious leaders for doing so. The thought was that people who were really "God people" wouldn't do such things and maybe Lydia reasoned that if she wasn't a real believer, then Paul, Silas, Timothy and Luke would not stay with her.)
One day, as they were going back to the riverside, they came into contact with a slave girl who was demon possessed and could predict the future. Because of her ability to tell the future, she made a lot of money for those who owned her. When she met Paul and the others, she followed them shouting, "These men are servants of the Most High God and they have come to tell you how to be saved." This went on for days until Paul became exasperated and turning to the girl, Paul spoke to the demon saying, "I command you in the name of Jesus Christ to come out of her." Instantly, the demon left her.
Obviously, that angered her owners because they would no longer be making good money off her fortune telling ability. In anger, they grabbed Paul and Silas and took them to the local authorities.
If you notice in verse 20, they don't accuse Paul of releasing the demon nor killing their fortune telling beusiness. Instead they screamed,"The whole city is in an uproar because of these Jews. They are teaching customs that are illegal for us Romans to practice." Following that, a mob formed against Paul and Silas and the officials ordered Paul and Silas to be stripped and beaten with wooden rods. After being severely beaten, they were thrown into prison. The jailer was ordered to not let them escape and he clamped their feet in stocks and put them in the inner dungeon.
This is so typical. Christians, and even Jesus, were brought before the authorities and charged with things not true. Their accusers never even spoke of the true reasons people had been brought before them. It seemed to always be about power or money but always expressed that the Christians were doing something anti-Roman. In reality, I imagine they could really care less about Roman law unless they could use it for their own good.
Around midnight Paul and Silas were praying and singing praises to God when there came a violent earthquake which opened all the prison doors and broke Paul and Silas from the stocks. The chains of every prisoner in the jail fell off as well. The jailer woke up to see the doors open and was preparing to jump on his sword to kill himself because he feared that he had failed and all the prisoners had escaped.
Paul shouted out, "Stop! Don't kill yourself! We are all here!" Hearing that, the jailer called for light to be brought in. He ran to the dungeon and fell down trembling before Paul and asked what he had to do to become saved. Paul told him all he needed to do was believe in the Lord Jesus and he would be saved along with everyone in his household. They shared the Good News with the jailer and all who were in his household.
The jailer took them all home, fed them, and washed and cared for their wounds. Then the jailer and everyone in his household believed and were immediately baptized. They all rejoiced and praised God.
The next morning the authorities sent word to the jailer to release Paul and Silas but Paul complained they were Roman citizens and had been beaten in public without a trial and if they wanted them to leave, the officials who did this would have to face them and release them themselves. When the officials heard they were Roman citizens they were alarmed and went to the jail and apologized and took them from the jail and begged them to leave. Paul and Silas went back to Lydia's home, encouraged the believers there, and then left.