Division- Paul and Barnabas have a sharp disagreement over taking John-Mark with them on a journey to encourage the new churches they just planted. They come to such an impasse that they must go their separate ways. Many have tried to determine who was at fault, but it seems that the most likely answer to that question is: No one.
Paul and Barnabas were two very different people with two different purposes and personalities. Barnabas was a man who continually put his own reputation on the line in order to bring those on the fringe with much potential for ministry into spiritual maturity. We saw him do this with Paul, now he is doing the same for John-Mark.
Paul is a man with a single-minded mission to preach the gospel to the ends of the earth. He can not risk mutiny in his mission team interfering with spreading the good news.
So their division is not sinful, but a natural result of two different people coming to a place where their mission and ministry requires that they part ways. And they were not enemies after this, in fact Paul speaks of Barnabas later with great respect, and even calls for John-Mark to come to his aid when he is imprisoned.
Addition- Because of their separation, Paul takes Silas, and Barnabas takes John, and they both set out to encourage the churches and continue to spread the gospel. There is an addition to the ministry teams that are at work. Now instead of one two-man team, there are now two two-man teams preaching and encouraging.
As he and Silas travel, Paul is impressed with Timothy, a young man in Lystra. They add him to their ministry team as well. Now, because of the addition of new members, the ministry teams have five preaching, teaching and encouraging.
In order for Timothy to be able to teach in the Synagogues, they circumcised him. This was not an issue of his salvation, simply an issue of conforming to cultural norms in order to reach those to whom God sent them to preach the gospel.
Multiplication- As the ministry team of Paul, Silas and Timothy (and in these verses Luke shows up with a "we" statement) are traveling to Asia, God turns them around and shows them that at this time the gospel is needed in Macedonia. God gives Paul a vision of a man from Macedonia crying out for help. They are obedient to go, and God rewards them with new converts, including a merchant woman named Lydia.
From these verses we learn that not every conflict is a result of sin. Sometimes issues arise because people need to move in two separate directions. As a result of Paul and Barnabas splitting up, they covered more ground and trained new leaders. We also learn that sometimes in ministering to those God has called us to, we must conform to aspects of their culture (that are not sinful in themselves) in order to reach them with the good news. Another lesson is that wherever we go, we know God is working there ahead of us to prepare the way. When Paul and his team came to Philippi, God directed them to Lydia who heard and responded to the gospel and then insisted that they allow her to provide hospitality for them while they were in the area.
As a result of the division, addition, and multiplication in these verses we have the foundation and background of four of the most influential New Testament writings: Galatians, Philippians, and I&II Timothy. And through it all we see God at work, building his kingdom and strengthening his church.