"The men were sent off and went down to Antioch, where they gathered the church together and delivered the letter. The people read it and were glad for its encouraging message. Judas and Silas, who themselves were prophets, said much to encourage and strengthen the brothers." NIV translation
So, the letter was delivered to the "church" in Antioch where it was received with gladness. Though it contained some restrictions on their freedom, they did not have to keep the entire Jewish law. That burden had been lifted and replaced by a smaller burden. The question is: What is the purpose of the Law? The answer is that it was given to make us (and them) see the need for a savior. We do not keep any part of the Law to be saved we try to follow the instructions of Jesus because we are saved.
"After spending some time there, they were sent off by the brothers with the blessing of peace to return to those who had sent them. But Paul and Barnabas remained in Antioch, where they and many others taught and preached the word of the Lord." NIV translation
So, Judas and Silas were sent back to Jerusalem after they delivered the letter. Paul and Barnabas stayed there, as it was their "home church". They taught and preached the Word to continue to strengthen the believers at Antioch.
"Some time later Paul said to Barnabas, 'Let us go back and visit the brothers in all the towns where we preached the word of the Lord and see how they are doing.'" NIV translation
After preaching/teaching in Antioch, it is decided that Paul and Barnabas will go and check on the groups of believers in the towns where they preached before. This is important in that they did not just leave new believers alone to figure things out on their own. They were going to disciple them, where it was needed, to help them to grow in their faith. This follows the pattern of what Jesus did with the apostles and it is the same pattern that we should follow today. We can see an example of what can happen when this pattern is not followed.
"Barnabas wanted to take John, also called Mark, with them, but Paul did not think it wise to take him, because he had deserted them in Pamphylia and had not continued with them in the work." NIV translation
This passage refers to the events of chapter 13 where we are told that John (Mark) left Paul and Barnabas and returned to Jerusalem. Because of that, Paul did not want him to go with them now. It is clear here that Paul did not feel that he could count on John Mark.
"They had such a sharp disagreement that they parted company. Barnabas took Mark and sailed for Cyprus, but Paul chose Silas and left, commended by the brothers to the grace of the Lord. He went through Syria and Cilicia, strengthening the churches." NIV translation
So, what started out as a seemingly minor matter ends in a disagreement that was large enough to break up the team. Barnabas took John Mark (his relative) and they went back home to Cyprus where they tended to the believers there. Paul chose Silas and they went back to the cities where they had shared the gospel and strengthened the "churches". There is good news in the relationship between Paul and John Mark as they are reconciled later (see 2 Timothy 4:11).