In order to establish his apostolic authority, and to demonstrate the supernatural origin of his message, Paul explained to the Galatians that, after his conversion, and after learning his ultimate mission, his first step was not to confer with the other apostles, learning from them what he should preach (Gal. 1.16-17a). Instead, he “went to Arabia, and returned again to Damascus” (17b).
It was not until “three years” later that he finally met with Peter and James in Jerusalem (vv. 18-19). Hence, he received his message “through the revelation of Jesus Christ,” not “from man” (Gal. 1.12).
Arabia, in the first century, encompassed a vast amount of territory. Some suppose it is
“probable that Paul's sojourn was in the desert bordering Syria, not far from Damascus” (Baker's, 216).
Intriguingly, Paul refers to Arabia again in this same letter as being the location of Mt. Sinai (Gal. 4.25). Perhaps the burgeoning apostle communed with the Lord at Sinai — as had several prophets before him — before returning to Damascus to preach in the synagogues of the Jews. Paul's mention of Arabia in Galatians 1 does seem to suggest that his purpose of going there was expressly not to “confer with flesh and blood,” but, in relative isolation, with the Lord (v.16).
At any rate, precisely where and why he “went to Arabia” after his conversion cannot be unequivocally determined.
The records of Saul's conversion in Acts 9, 22, and 26 do not mention his journey to Arabia. It is possible it transpired somewhere between Acts 9, verses nineteen and twenty, or, as others suppose, between verses twenty-two and twenty-three.