It is an interesting exercise to compare the autobiographical anecdotes in Paul’s letters about his call to proclaim the Messiah among the nations and the ensuing debate about how such a mission was to be conducted with the later account of these same events in the book of Acts. According to Paul, he initially responded to his prophetic calling by going to Arabia before he returned to Damascus (Galatians 1:17), while Acts 9:19-22 follows the narrative of the vision that blinded Paul on the road to Damascus with his subsequent ministry in Damascus. Both 2 Corinthians 11:32-33 and Acts 9:25 agree that Paul had to secretly escape from Damascus by being lowered in a basket over the city’s wall, though the former claims that the threat came from Aretas IV, the king of Nabatea (ca. 8 BCE – 40 CE), and the latter that the threat was from a local Jewish group that was antagonistic towards Paul’s message. Acts 9:26-27 then describes how Barnabas introduced Paul to the “apostles” in Jerusalem, whereas Paul recounts his first meeting with Peter and Jesus’ brother James as taking place three years after his “call” (Gal 1:18).
Here is where it gets more complicated. Paul goes on to Caesarea and then Tarsus (Acts 9:30) when his storyline is interrupted to narrate a few stories about Peter’s miracles and his preaching to the first Gentile “converts” (9:32-11:18). The spotlight shifts back to Paul in Acts 11:25 as Barnabas finds Paul in Tarsus and brings him to Antioch where the pair have a year-long ministry. In Antioch, a prophet named Agabus predicts a famine in Judaea, which the narrator dates to the reign of the Roman emperor Claudius (ca. 41-54 CE), and Paul and Barnabas are sent to Jerusalem with resources to assist with the famine relief (Acts 11:27-30). This seems to be Paul’s second visit to Jerusalem (but cf. Mark Goodacre’s case that we have two variant accounts of one meeting). Acts 13-14 covers Paul’s first missionary journey which terminates when some individuals from Judaea demand that Paul’s non-Jewish Christ-followers get circumcised, leading Paul and Barnabas to travel to Jerusalem a third time to resolve this matter in a public conference with the apostles and James in Jerusalem (Acts 15:1-29). Unfortunately, although the side of Paul and Barnabas was vindicated at the Jerusalem Council, the two missionaries part ways in Antioch (15:35-41).
Galatians 2:1-10 may be describing the last meeting in Jerusalem between Paul, Barnabas, and the Jerusalem Pillars over the question of whether non-Jewish Christ followers had to fully adopt Jewish customs including circumcision and Torah-observance. Moreover, Paul follows this report by sharing a conflict over table fellowship between Jews and Gentiles that he had with Peter and Barnabas in Antioch (Galatians 2:11-14). However, where Galatians seems to differ is that Paul does not speak about any other visit to Jerusalem between his first meeting with a few key church leaders there and his private conference with them 14 years later and Paul does not give any details in Galatians about the Jerusalem Decree (Acts 15:20-21, 29) that apparently resolved the issue of how non-Jewish and Jewish Christ followers could maintain fellowship with each other. Thus, some scholars argue that Paul and Barnabas raised the concerns that they had in Galatians 2:1-10 during their second visit with the apostles/Pillars in Acts 11:27-30 and that the controversy in Antioch in Galatians 2:10-14 actually precipitated the great Jerusalem Conference in Acts 15. On the other hand, other scholars respond that there are far more correspondences between Galatians 2:1-14 and Acts 15 than with Acts 11 and it is unlikely that 14 years transpired between Acts 9:26 and 11:27.
Here are some online articles and blog posts by Pierson Parker, Robert H. Stein, Ben Witherington III, Mark Goodacre, Matt Page, J. Peter Bercovitz, Loren Rosson, Bill Heroman, Phillip J. Long, Jeremy Sweets, and Michael Barber that discuss this chronological issue in further detail. Please let me know if there are other blog posts on this question. My own inclination is that the author of Acts has deliberately reshaped the account of Galatians 2 to suit Acts’ larger theological purposes.