* Harry Potter was the first to suspect the monster was a Basilisk. He recognized the symptoms of petrification as being caused by a Basilisk's gaze, and he even saw the creature's shed skin. However, he didn't fully confirm its identity until he saw it himself in the Chamber of Secrets.
* Professor Dumbledore was likely the first to know for sure that the monster was a Basilisk. He was the only one who knew about the Chamber's existence and the creature's history, and he likely had access to information about Basilisk behavior.
* Professor Lockhart claimed to have "found out" about the Basilisk, but as we know, he was a fraud and had no real knowledge of the creature.
So, depending on what you consider "finding out," the answer could be Harry, Dumbledore, or even the entire wizarding world, as the truth was eventually revealed to everyone.