The persecution of the church that began with the stoning of Stephen results in the scattering of much of the Church. While the Apostles stay in Jerusalem and continue their work there, many Christians leave Jerusalem and scatter to the country side. This scattering has the positive result of sending missionaries into places the gospel had not yet reached. The old ones used to say "the blood of the martyrs is the seed of the church." Under persecution the Christian movement has always gone deep and has thrived. Historically, the church has never survived complacency and ease.
One consequence of this scattering is some Cypriots have gone to Antioch and converted a number of Greeks. This revival in Antioch gets the Apostles attention and they send Barnabas to check things out. Barnabas real name is Joseph of Cyprus (note Joseph is from Cyprus is you are wondering why the Apostles chose to send him). The Apostles called Joseph "Barnabas" because of his ministry and temperament. The names means "Son of Encouragement." Great name!
Barnabas surveys the situation and realizes that he needs a teacher to build up the Church. He goes to Tarsus to find Saul -- you'll remember that when Saul is converted on the road to Damascus he makes his way to Jerusalem but because of his reputation as a persecutor of Christians cannot get an audience with the Apostles. It is Barnabas who takes Saul and introduces him around (again encouraging!). Barnabas brings Saul to Antioch and they spend a year teaching and preaching and building up the church there.
It is in Antioch that the Christ followers are first called "Christians." Scholars believe that this was originally a term of derision. It was meant as an insult. Clearly the early movement embraced the insult and chose to be known as "Christers!". Early Methodism has the same history. The name "Methodist" was supposed to be an insult hurled at John Wesley's Oxford "Holy Club" because they were teaching a method of spiritual growth. Wesley, being the wise leader he was, embraced the insult and called his group "Methodist!".