Which anime story came first, Accel World or Sword Art Online? I saw over the internet that Accel World came out sometime in 2009 and Sword Art Online came out around in 2013, but it seems like many people saying SAO came first than Accel World.

Which came first, Accel World or Sword Art Online?
Accel World came first. According to their Wikipedia pages, Accel World light novel started in February 2009 while Sword Art Online started in April 2009. If you are referring to anime release, then the same can also be said since Accel World anime started airing in April 2012 while SAO anime started airing in July 2012. 

As pointed out by @Memor-X, some people say that SAO came first but this most likely refers to the chronology of the events in both stories and not to the release order of light novels/anime because as mentioned here, the events in SAO happened before the events in Accel World on the same universe/timeline.