What was the first set produced by The LEGO Group?
Some of the earliest "LEGO brick" sets I could find were these three on BrickLink from 1954:





As other answers have noted, there were earlier sets than this, but they were referred to as "Automatic Binding bricks". It wasn't until 1953 that the bricks were officially renamed as "LEGO bricks" (i.e. "LEGO Mursten").

These sets were using precursor designs of the "modern" plastic LEGO brick without the hollow underside tube. It wasn't until 1958 that the brick design we all know and love today was created. It was officially "born" at exactly 1:58 p.m., January 28, 1958, when it was first patented.

So, if you're looking for the earliest official "LEGO bricks" using the modern stud-and-tube design, these are the sets you're looking for: