Where was Kirito running to?
According to the first light novel he was heading to a weapons shop.


  After I had logged onto SAO, I started running through the
  nostalgic stone paved roads of the «Starting City», headed for the
  weapons shop. Realizing that I was a beta tester after seeing me start
  and dash, Klein ran over without any hesitation.
  from chapter 2 of the first volume