"Now those who had been scattered by the persecution in connection with Stephen traveled as far as Phoenicia, Cyprus and Antioch, telling the message only to Jews." NIV translation
This verse refers back to the events recorded in chapters seven & eight. After Stephen was stoned to death, persecution broke out in Jerusalem and many followers of Jesus Christ fled from the city. These people took the message to the Jewish people in these other places. God used this persecution to get the people out of their "comfort zones" and to spread the gospel. They were still stuck on the tradition of God belonging to the Jewish people and so they only shared the message with Jews.
"Some of them, however, men from Cyprus and Cyrene, went to Antioch and began to speak to Greeks also, telling them the good news about the Lord Jesus. The Lord's hand was with them, and a great number of people believed and turned to the Lord." NIV translation
At this point in time, Antioch was the capitol of Syria and a very important city.
Some of the men that fled the persecution in Jerusalem stopped here and began to share the gospel with Greeks as well as Jews. These men would have been considered radicals as they went against tradition and shared the gospel with non-Jews but with God all things are possible and a great number of people were saved.
"News of this reached the ears of the church at Jerusalem, and they sent Barnabas to Antioch. When he arrived and saw the evidence of the grace of God, he was glad and encouraged them all to remain true to the Lord with all their hearts. He was a good man, full of the Holy Spirit and faith, and a great number of people were brought to the Lord." NIV translation
A great number of people were coming to the Lord and news of God working in this city spread. When the news reached Jerusalem, the church leaders dispatched Barnabas to check it out. In this passage, we see that Barnabas could tell that it was a work of God "evidence of the grace of God". That should be a mark of everything we do as Christians. Those that see what we do should be able to see that it is God at work and not of us. Barnabas saw this in Antioch and was pleased to encourage them in the faith. As Christians, we are also called to celebrate and encourage our brothers and sisters.
"Then Barnabas went to Tarsus to look for saul, and when he found him, he brought him to Antioch. So for a whole year Barnabas and Saul met with the church and taught great numbers of people. The disciples were called Christians first at Antioch." NIV translation
As great numbers of people were coming to Christ, Barnabas realized that he would need help in training up these new believers so he went to get Saul. They spent a year in the city teaching the people who were known as Christians (followers of Christ).