Timothy: Derby and Lystra (16:1-5)
The reason why Paul had Timothy circumcised and not Titus is given to us; "because of the Jews which were in those quarters: for they knew all that his father was a Greek" (Acts 16:3). The issue with Timothy was that one parent was Jewish and the other Gentile, and so it would raise distracting questions about his ethnicity among the Jews. Paul used his Christian liberty, so that "unto the Jews I became as a Jew, that I might gain the Jews; to them that are under the law, as under the law, that I might gain them that are under the law" (1 Cor. 9:20). Paul did not compromise his principles with Timothy, but had rather used his Christian liberty to bring an end to any distraction over the issue. In Acts 15 it was a false teaching coming in among believers that circumcision was required for salvation. In Acts 16 it was to remove confusion among unbelievers whom Paul was trying to reach with the gospel. What the Jews wanted was to reject Timothy because of his mixed background, so Paul had him circumcised! The case of Titus was different (Gal. 2:3). Both the parents of Titus were Gentiles, and therefore he did not pose the same issue. In fact, the circumstances of him being a Gentile and uncircumcised is connected with why the Lord revealed to Paul that he should take Titus with him to the Jerusalem council. Titus was not compelled to take up with Jewish ceremony. To do so would be a compromise of principles! This makes Paul's position very clear.
Directed to Macedonia (16:6-10)
In scripture there is a difference between "believing", "believing in", and "believing on". If you "believe" someone, you accept what they say as truth. If you "believe in" someone, you have confidence in their character; i.e. it has to do with who they are, although it also includes that they are honest. If you "believe on" someone, they become an object for your faith, and this also includes both confidence in their character and in their words, but goes far higher.