Consider that basically, a bildungsroman is a coming-of-age novel. Growth and development of the main character in such a novel is often focused on the major life lessons the character learns, and how he or she changes as a result. Whether formal or not, this type of novel focuses on the education of the main character.
Great Expectations follows its main character, Pip, from the time he is a poor 7 year old boy in a small English village, until he has grown into a young man with a mysterious benefactor funding his becoming a "gentleman" in the city of London, until finally, through different trials and hardships, he returns once again to his village.
We watch as Pip is educated in the way of social acceptance into a world that has expendable funds. At the same time, he is conflicted over feelings for a girl from his childhood. While he never fully loses himself or his roots, when a visit from Joe brings him shame he realizes that the man he is becoming is not the boy he once was. He is taken on a series of life adventures, some which eventually lead him to his benefactor, others which enable him to becomes a businessman.
By the end of the novel, Pip comes full circle, returning to his village once again equal to those who live there. Through everything, he learns who loves him most in life, and decides to to be devoted to those relationships which have always been devoted to him.
You have identified a vital element in this novel. It is clearly a bildungsroman, just like Jane Eyre and other similar novels, in that it traces the development of a main character from their youth and to their maturity. At the end of bildungsromans, the characters have normally gone through some hard times but have found their place in society and end up a maturer, wiser individual because of what they have suffered.
Thus in analysing this novel as a bildungsroman, it is important to note how Pip changes and develops. Key to the early stages of the novel is Pip´s growing sense of class consciousness and his dissatisfaction with his own position in society. We can see that this comes through his first meeting with Miss Havisham and Estella, and how he becomes ashamed of his humble roots. The end of Chapter 9 clearly marks this event as a fundamental point of change in his life.
That was a memorable day to me, for it made great changes in me. But it is the same with any life. Imagine one selected day struck out of it, and think how different its course would hgave been. Pause you who read this, and think for a moment of the long chain of iron or gold, of thorns or flowers, that would never have bound you, but for the formation of the first link on one memorable day.
This passage is key because it introduced the notion of being "bound" - something that is developed in Ch. 13 when Pip is "bound" into his apprenticeship. This indicates how Pip sees his lowly position. It is important to note that this awareness or realisation comes before Pip receives his "Great Exepectations", and so it is likely that if Pip had not received his Great Expectations he would have lived a frustrating and sad life, consciously aware of the limitations of his position in society and unable to do anything about it. With his visit to Estella, gone is the ability to accept his fate.
Moving forward to the rest of the novel, it is key to identify the retrospective omniscient voice that narrates Pip's tale, and how it demonstrates how Pip has matured and become wiser. Certainly the wisdom that the narrator demonstrates has only come through the sufferings and trials that the younger Pip experiences - there is a definite sense that this is a novel of maturing, of change and growth in character. The incident where Pip saves Miss Havisham from being burnt and also burns himself in the process, and his loss of his "expectations" and the fever that cripples him have a sense of purgatorial repayment for the wrongs that Pip has committed - he learns just how much of a snob he has been, and how he has hurt others through his actions, and begins to right his wrongs. Thus we see at the end of the novel a sadder, but much wiser Pip, who has definitely learnt a lot through his experiences, and has found his place in society working with Herbert.
It is surely significant that at the end of the novel, Joe and Biddy call their son Pip after the main character of the novel. After reading a novel which is so much concerned about parental figures and the kind of (bad) influence they can have on their "children" (just think of Miss Havisham and Estella and Magwitch and Pip), we can only think that this naming will have a good impact, as the younger Pip now has an appropriate role model to follow in his life - one that can guide him with sensitivity and wisdom.
Much like the monologue of the melancholy Jacques from Shakespeare's As You Like It, known as "The Seven Ages of Man," Great Expectations chronicles in prose, rather than poetry, the three stages of Pip from somewhat lonely and naive orphan who is made aware that he is a "common, a mere labouring-boy," who forms a link to his life on a single, memorable day of "great expectations" to a young gentleman now recklessly foolish, rejecting his old friend--
All other swindlers upon earth are nothing to the self-swindlers, and with such pretenses did I cheat myself.
--to that of a man who has reached a maturity of soul, having weathered failure and loss of friendships, loss of love, and loss of friends in a final stage of rebirth and re-creation. Pip apologizes to his friends Joe and Biddy, and returns to the warmth of the forge. Joe marries Biddy, and Pip visits. Then, he meets with Estella, closing relationship with her. In the final stage, Pip comes full circle and learns that he has left his home in search of illusionary "great expectations" when the best values in life have always been there on the forge. Pip writes to Biddy,
"My dear Biddy, I have forgotten nothing in my life that ever had a foremost place there and little that ever had any place there. But that poor dream, a I once used to call it, has all gone by ,Biddy, all gone by!"