All of the lessons suggested thus far in the answers are quality ones from the book. I'd like to suggest one that revolves around one of the title characters, the Half-Blood Prince.
One of the central mysteries of Rowling's book is the identity of the Half-Blood Prince. Harry Potter finds a used potions book with all sorts of useful instructions in it. A name, the Half-Blood Prince, is written inside it. This book gets Harry through his potions class and also gets him into some serious trouble. Potter tries a spell that is written in the book and ends up seriously hurting another student.
But, even despite the danger, Potter has a deep connection with the book. He feels that the Half-Blood Prince is a lot like himself. He hides the book from authorities within in the school, hoping to keep it for himself.
However, at the end of the book, Potter finds out the true identity of the Half-Blood Prince.
Just after Snape has killed Dumbledore, Potter tries to use one of the spells from the book on Snape, who easily counters the spell. Snape reveals that he is the Half-Blood Prince.
So what is the moral lesson from all this? I think Rowling is warning against things that seem too good to be true. All throughout the book Potter is warned about how dangerous the Half-Blood Prince's potions book might be. His closest friends continually warn him that he's getting into trouble. But Potter holds on to the book because he likes it.
And, in the end, this turns out badly for him. Rowling is warning us to trust our friends and to have our eyes open about things that seem too good to be true. Do look a gift horse in the mouth, she says.
There are lots of themes and lessons in this book.
To me, perhaps the biggest lesson is that murdering people in cold blood tears your soul apart (at least in Rowling's opinion.) This is the book where we learn about Horcruxes, and I think that is the message of the Horcruxes.
I think there are some other themes. First, I think we see the theme of self-sacrifice. Both Harry and Dumbledore are doing a lot of stuff that they would rather not have to do, but they are going to do it because it is necessary. Dumbledore, for example, drinks the potion at the lake to get the locket. And then he allows himself to be killed. Harry sacrifices time with his friends and his fun to go through this all with Dumbledore.
You can also see the importance of true trust and friendship with the whole deal between Lavender and Ron and Hermione. And we see Harry being a friend by backing off of doing anything about liking Ginny for a long time because he doesn't want to hurt his relationship with Ron.
I don't think that this novel contains as many lessons as in the Deathly Hallows, where Rowling explains through Dumbledore's words the moral of the entire series.
One lesson, I think, would be sympathy even for one's enemies. Though Draco Malfoy is a Death Eater who plotted to kill Dumbledore, Rowling potrays him sympathetically in several scenes. In one scene, he is crying due to worry for his family, and he is unable to actually kill Dumbledore. This shows that Draco has not yet been fully corrupted. I think that this ties in with Rowling's theme that the world is not morally black and white. Draco, who would appear to be an "evil" character, is clearly morally ambiguous. He also has a good, still innocent side to him.
Harry Potter and the Half Blood Prince has many themes, morals, and lessons depending on the angle you look at it from. The novel has a side story of Draco's coming of age. He is forced to make a difficult choice that isn't really a choice at: become a death eater or die with his parents. So, one of the most important lessons from this novel is not to judge people too harshly because its impossible to know what they are going through.