One of the morals of the story is that you never know what you are capable of until you try. Harry was faced with extraordinary circumstances, and he was able to rise to the challenge.  He found reserves of bravery within himself, as well as special abilities.

Harry Potter had no idea that he was a wizard, of course.  When he was eleven years old, he found out he was a wizard and was whisked off to wizarding school.  That was pretty unusual.  Until then, he had no idea he was “The Boy Who Lived” or that he had famous parents.

From the beginning, Harry exhibited traits of bravery and curiosity.  He saw Hagrid take something from a vault at Gringotts, and he wanted to protect it.  He thought that it was Snape who was after it, and about that he was wrong, but at least he tried to protect Hogwarts.  Harry made friends based on their character traits and not prestige.  He discovered that he had talents he never knew he had, including an aptitude to play the sport Quidditch.

Harry risked his life to follow the Sorceror’s Stone deep into the castle to stop anyone from taking it.  He thought it was Snape, but it turned out to be Quirrell. Harry was brave enough to fight off Quirrell, even though he was harboring Voldemort.

Harry jumped to his feet, caught Quirrell by the arm, and hung on as tight as he could. Quirrell screamed and tried to throw Harry off – the pain in Harry's head was building -- he couldn't see -- he could only hear Quirrell's terrible shrieks and Voldemort's yells of, "KILL HIM!... (Ch. 17) 

Harry is later told by Dumbledore that he saved the stone because he wanted to rescue it, and not use it.  In this way, the person who took the stone had to have good intentions. Harry had good intentions, whereas Quirrell and Voldemort did not. 

You see, only one who wanted to find the Stone -- find it, but not use it -- would be able to get it, otherwise they'd just see themselves making gold or drinking Elixir of Life. (Ch. 17) 

Voldemort got away, but he did not get the stone.  This was directly a result of Harry’s bravery.  He found that he had many traits he did not realize he had before he went off to wizarding school.

What are the moral lessons in Harry Potter and the Sorcerer's Stone? 

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.