Did Barnabas go looking for Saul or did he just accidentally run into him?

Barnabas deliberately went looking for Saul; it was not an accidental meeting. He sought Saul out, found him, and brought him back to Antioch, where Paul and Barnabas then ministered to that Gentile congregation for 18 months. The context shows an intentional search, not a chance encounter.

Les teaches that Barnabas was God's man at the right time, and when he saw that God was beginning to deal with Gentiles, Barnabas went looking for Saul. He did not just accidentally run into him; he intentionally went looking for him. When he found him, he brought him back to Antioch, and Paul and Barnabas ministered there to that Gentile congregation for 18 months. Les then notes that chapter 12 transitions away from Paul and Barnabas for a little while and returns to Jerusalem and Peter.

When did Barnabas go looking for Saul?

Barnabas went looking for Saul at the right time, when he saw that God was beginning to deal with Gentiles. It was not accidental; he intentionally sought him out, found him, and brought him back to Antioch, where they ministered to the Gentile congregation for eighteen months.