Some believe the "many days" referred to here encompasses the three years that Saul spent collectively in Damascus, Arabia, and then back in Damascus (before ever going to Jerusalem as a Christian; cf. Gal. 1:17,18). It was likely during this time that Saul received revelations from Christ that determined his theology for him (cf. Gal. 1:11,12). If this chronology is correct, then it took many months or even several years before the first attempt was made by the Jews against Saul's life. They could only put up with him confounding them for so long (as was also the case with Jesus and Stephen)!
Regardless of the timing, the Jews who do not believe in Jesus desire strongly to do away with Saul and his testimony. The ill treatment he had directed against Christians was now directed against him. The Jews watch the city gates continuously, believing they will catch him when he attempts to leave Damascus. However, the disciples sneak him out through a hole in the wall by means of a large basket (cf. II Cor. 11:32,33).
"And when Saul had come to Jerusalem, he tried to join the disciples; but they were all afraid of him, and did not believe that he was a disciple. But Barnabas took him and brought him to the apostles. And he declared to them how he had seen the Lord on the road, and that He had spoken to him, and how he had preached boldly at Damascus in the name of Jesus. So he was with them at Jerusalem, coming in and going out. And he spoke boldly in the name of the Lord Jesus and disputed against the Hellenists, but they attempted to kill him. When the brethren found out, they brought him down to Caesarea and sent him out to Tarsus" (Acts 9:26-30).
It is difficult to imagine the powerful emotions that must have swept over Saul as he returned to Jerusalem after such a span of time and under such different circumstances than when he last left. Seeing the city gates again, the temple, the spot where he assisted in Stephen's stoning, the faces of his old allies, and family members of Christians he had persecuted (some to the point of death) would have weighed heavily upon anyone, even a strong man like Saul.
Saul's initial attempt to identify himself with the Christians in Jerusalem was not successful. They believed he was trying to deceive them. It took the confidence and encouragement of Barnabas to convince the other Christians to embrace Saul as a genuine disciple. The kindness and trust shown by Barnabas certainly helped create the bond that would eventually blossom into a wonderful working relationship between these two men (cf. 11:25ff). After the church accepted Saul (which would have been quite a challenge for those who may have lost loved ones as a result of Saul's prior activities), he labored with the disciples there and preached the gospel boldly, certainly winning over any followers of Christ who still doubted him. He infuriated some Jews to such a degree, however, that they made an attempt on his life. It was unsuccessful and the Christians wisely moved Saul to a region they believed would be safer for him (which was in harmony with the Lord's will; cf. 22:17-21). It should be noted that there are legitimate ways of avoiding persecution (sometimes relocating is the best course of action). It is at this point that Luke temporarily stops detailing Saul's activity and turns his attention to the labors of the apostle Peter, after a transitional verse.
"Then the churches throughout all Judea, Galilee, and Samaria had peace and were edified. And walking in the fear of the Lord and in the comfort of the Holy Spirit, they were multiplied" (Acts 9:31).
This is a wonderful summary statement regarding the progress of the gospel (from Jerusalem to all Judea, Galilee, and Samaria) and the unity and strength of the congregations of Christ's body (they were at peace and being built up). To the extent that they continued trusting God, fearing Him, and seeking comfort from the indwelling Spirit and His word, God would continue to bless them with growth.