And when Saul was come to Jerusalem, he assayed to join himself to the disciples: but they were all afraid of him, and believed not that he was a disciple.
Note 1 at Ac 9:26: This verse leaves the impression that Paul went immediately from Damascus to Jerusalem. However, in Ga 1:15-18, Paul implied that three years passed after his conversion before he went to Jerusalem and spent fifteen days with the Apostle Peter.
It is possible that during the "many days" of Ac 9:23, Paul could have gone into Arabia and then have returned to Damascus, but the Jerusalem disciples' fear of him makes it look like Saul's appearance in Jerusalem was shortly after his conversion. If this would have taken place three years later, it seems most Christians would have viewed his noticeable lack of persecution of the saints as proof enough that his conversion was genuine.
It is also possible that Ga 1:18 and this account are describing two separate trips to Jerusalem. Saul could have gone to Jerusalem immediately after leaving Damascus and then returned three years later and spent fifteen days with the Apostle Peter. The statement in Ga 1:17, "Neither went I up to Jerusalem to them which were apostles before me," is emphasizing the fact that Paul was not taught his revelation of the New Testament by any of the apostles; it does not rule out that he had been in Jerusalem since his conversion.
Another explanation is that Paul spent three years between Damascus and Arabia then went to Jerusalem where he spent fifteen days with Peter (Ga 1:18).
Note 2 at Ac 9:26: The believers weren't sure Saul had really been converted. They thought this might be a trick so that Saul could obtain names that would aid in his persecution of the saints. However, Ac 9:27 shows that after Barnabas explained what had happened at Damascus, they did receive him without restrictions. This showed a great faith on the saints' part.