Pip learns that wealth does not ensure that one is a true gentlemen. Look at the wealthy people in the novel - they are all miserable people (Miss Havisham, Estella, Jaggers, Uncle Pumblechook, etc.) the truly good people are those like Joe, Biddy and Herbert Pocket that love Pip for who he is, not what he is expected to become.
He also learns that life sometimes can be ironic. All the while he thinks that Miss Havisham is his benefactor, and it turns out to be a convict.
He learns that one good turn deserves another. He was kind to Magwitch way back when he was a child, and Magwitch rewards him with his inheritance later in life.
He learns that there is no justice in this world. Magwitch, although a criminal, has a good heart and in many cases, has been unjustly convicted.
He learns that vengeance is destructive. Miss Havisham spends her life trying to avenge herself against a man who jilted her, and it destroys not only her, but Estella, because Miss Havisham has used Estella as an instrument of her vengeance.
There are many other lessons - check out the "themes" section here on eNotes.
As readers follow Pip over the course of over twenty years, they see him experience and learn many things. I think the most important lesson, though, is one which takes Pip much of the novel to understand.
In Great Expectations, Dickens makes clear--and even exaggerates--the differences in social class in Pip's society. As the novel opens, readers meet a young Pip, who is being raised by his sister and her husband, Joe, a blacksmith. As Pip does not know any other way of life, he does not see anything wrong with the life his family leads. However, as soon as he meets Estella, he becomes aware of her beauty and sophistication--characteristics that drive Estella to point out Pip's "commonness."
When Pip leaves Miss Havisham's for the first time, he feels his own perception of himself is permanently changed:
I set off on the four-mile walk to our forge, wondering, as I went along, on all I had seen, and deeply revolving that I was a common labouring-boy; that my hands were coarse; that my boots were thick; that I had fallen into a despicable habit of calling knaves jacks; that I was much more ignorant than I had considered myself last night; and generally that I was in a low-lived bad way.
From this point on, Pip's obsession with "elevating" himself from his social class becomes central in the novel, as Pip feels that achieving this goal will get him Estella.
As lynnebh notes, most of the characters with whom Pip desires to keep company are not at all deserving of such attention. Further, Pip alienates and disrespects good people like Joe and Biddy, and Pip's utter disgust upon learning that the convict, not Miss Havisham, is his benefactor marks one of the most heartbreaking scenes in the novel to readers.
(As an aside, it's important to note ways in which Dickens creates exaggerated caricatures to show the ridiculousness of people's obsession with money and the value they place in it. Miss Havisham is a complete mess, but her money brings her attention from family members and nosy neighbors who want to know her. Similarly, Mrs. Pocket, a woman who is obsessed with "titles" and claims to be "almost" royalty, is completely and utterly incapable of performing the most basic tasks, such as looking after her own children.)
It isn't until close to the end of the novel that Pip finally realizes the implications of his own behavior. He recognizes that he has wronged many people, and finally learns how to love people no matter what their social status is. And finally, he realizes that a person's social status does not dictate his or her worth as a person.