Acts 16 is a turning point for a number of reasons. From now on, Acts will mostly tell the story of Paul and his companions. We see the Church expand its reach into a new continent (Europe), and new characters are introduced, including Timothy and - apparently - Luke himself.
The hint is a shift in pronouns, from "they" to "we." It occurs in vs. 10. It is believed that Luke enters the story in the city of Troas, joining Paul and his companions just as they are about to leave Asia and enter Europe for the first time.
We also meet Timothy in vs. 1, the son of a Jewish-Christian woman and Gentile father. Timothy will become Paul's protege. The question may be asked why was Timothy circumcised in vs. 3 when the Church had just rendered its verdict that such surgery was not necessary for salvation. The answer appears to be that while circumcision isn't required to be a Christian, it still proved useful and advantageous in gaining the Gospel a hearing among die-hard Jews. Jewish resistance to associating with Gentiles was so strong that, for missional purposes, Timothy agreed to become like the Jews in this regard rather than be a stumbling block for them. (See the Acts 21 riot for a picture of how hostile Jews could be to the uncircumcised!) I would view this act as a cultural concession, not a spiritual compromise.
Just like we have books of the Bible named after Timothy and Luke, so, too, do we get introduced to some new geographical locations in Acts 16 that will become the recipients of biblical epistles: Galatia (vs. 6) and Philippi (vs. 12) .
Several interesting encounters occur in Philippi. It is a largely Gentile city (remember we are getting farther away from the Holy Land and its outlying Jewish settlements), so much so that there doesn't appear to be an established synagogue for the Jews, but merely a place by the river where they would meet on the Sabbath. Lydia - a woman - becomes the first Christian convert in Europe.
A spirit-possessed slave girl begins following Paul and his companions around, testifying to their mission. Just like Jesus in the Gospels didn't care to have the testimony of demons on His behalf, so, too, is Paul more irked than pleased by this slave girl. He exorcises the spirit from her - which provokes legal problems for him and his team. As usual, the disciples get into the most trouble when there is an economic cause. In this case, the removal of the slave girl's spirit of divination meant a financial loss for her owners. They stir up the town against Paul and Silas and have them beaten and jailed.
The story of the Philippian jail is a beautiful account of Christian grace under pressure: Paul and Silas singing and praising God while bruised and bleeding, and refusing to escape when given the opportunity (to do so would have put the jailer's life in jeopardy; he was preparing to commit suicide because -as a jailer - his life would have been forfeit had his prisoners escaped. The prevailing view would be that such an instance meant the jailer had been bribed.) The jailer is so impressed by the witness of the apostles that he and his household become believers and are baptized that night. We also discover during this episode that Paul is a Roman citizen, a fact which will come in handy during his adventures throughout the Empire.
Definitely, the Church is on the move!