The main lesson in the story is that you should not judge people based on appearances. By this I mean what people look like but also their actions. Sometimes people have secrets that you just do not expect. Harry thought that Snape was out to get him because he treated him unfairly and he seemed odd. In reality, the quivering Quirrell was hiding Voldemort.
Harry felt that Snape was targeting him because he was a dark wizard. Snape’s classroom was in the basement and he certainly wasn’t winning any personality contests, but this led Harry to make the assumption that Snape was trying to steal the Sorcerer’s Stone. He didn’t look closer to see what was really happening.
Quirrell seemed completely harmless. When the children were trying to stop Snape from cursing Harry’s broom during the Quidditch match, they actually managed to stop Quirrell instead.
Hermione had fought her way across to the stand where Snape stood, and was now racing along the row behind him; she didn't even stop to say sorry as she knocked Professor Quirrell headfirst into the row in front. (Ch. 11)
The children do not realize until it is too late that Snape has been trying to protect Harry and Voldemort is trying to get the Stone from Quirrell, the actual Death Eater. Harry is able to prevent Quirrell from getting the stone because Dumbledore provided an enchantment so that only someone who did not intend to use it could possess it.
Harry has a good heart, but he does not always think things through. He needs to learn that appearances can be deceiving and you should not judge people based on limited information. He does realize this, as he makes friends with two of the most unusual children at Hogwarts. Hermione seems like a know-it-all and Ron Weasley is the youngest son of a poor family. Harry is able to see past things that might make them unpopular to see that they are good friend material.