Why did Paul go to Arabia? In the first chapter of Galatians, paul said,


  17 Neither went I up to Jerusalem to them which were apostles before me; but I went into Arabia, and returned again unto Damascus.


After he his encounter, he should have gone to Rome or Jerusalem but he went to Arabia. Does mount Sinai has any importance since law was given on it so may be he went there?
Galatians 1:15-18 (ESV)
  
  But when he who had set me apart before I was born, and who called me
  by his grace, was pleased to reveal his Son to me, in order that I
  might preach him among the Gentiles, I did not immediately consult
  with anyone; nor did I go up to Jerusalem to those who were apostles
  before me, but I went away into Arabia, and returned again to
  Damascus. Then after three years I went up to Jerusalem to visit
  Cephas and remained with him fifteen days.


It is not clear why Paul went to Arabia. Even Luke did not mention this in Acts. We can find this account only in his letter to Galatians. We may conclude that Paul stayed sometime in Damascus after the conversion, preaching to the Jews and then went to Arabia, and after three years only Paul went to Jerusalem to speak with the Apostles face to face.

Some scholars believe that Paul went to Arabia to receive revelations from Jesus Christ.


  From the reading of the text in Acts 9:26-28 one would get the
  impression that Saul went directly from his escape at Damascus to
  Jerusalem. By his own admission he clarified the fact in his letter to
  the Galatians (1:16-17) that he “conferred not with flesh and blood”
  nor consulted with the Apostles in Jerusalem; but the Lord’s choice
  for him was to go to Arabia to be trained in the school of the Spirit
  in order that he might receive greater revelations concerning the
  mysteries of the Gospel of the glorified Christ. (source)


Some scholars believe that Paul went to Arabia to preach the Gospel and this action proves that Paul received his revelations concerning the Gospel from Christ himself.


  Paul thus wrote to tell them that after his conversion, he preached
  among the Gentiles for an extended amount of time before ever meeting
  with another apostle. Paul did not hurry off to Jerusalem to get
  instruction and approval from the Twelve. In defense of his apostolic
  credentials to the churches of Galatia, Paul mentioned his delayed
  journey to Jerusalem in order to emphasize (among other things) his
  genuine apostleship, whose message and authority came from Almighty
  God, and not from the twelve apostles, or any other person.
  (source)


Some again believe that Paul needed solitude because his life suddenly turned upside down.


  Another point to consider is Paul’s world was turned upside-down in a
  moment’s time. Wasn’t he human, just like the rest of us? Wouldn’t he
  need to withdraw for awhile in solitude to more or less come to grips
  with the powerful upheaval that had disturbed the certainty of his
  righteous stand in the Law? Wouldn’t he have needed some inner
  clarity? Up to this point in his life, he had served the Law, and
  through the Law, he was convinced that the new Messianic Jews taught
  blasphemy when preaching that the Temple sacrifices had become
  obsolete in Christ. (source)