Elizabeth begins as a character that is indepedent and quite proud in her assessments of people. It is her pride that is injured when she overhears his first comment at the party, about her not being particularly pleasant or beautiful, and that wounded pride that keeps her from allowing Darcy to win her heart, and that pride leads to her cruel rejection of his marriage proposal. However, a series of events humbles her pride, and teaches her that she is no better than Darcy, who is in fact a wonderful person. The first event is the letter that writes to her explaining his connections with Wickham and his motivations for discouraging Bingley's attentions to Jane. The letter is so sincere, genuine and truthful that Elizabeth is ashamed of herself. Next, when her sister Lydia runs away with Wickham, she realizes that Darcy might have been correct in his assessment of her family--that humbles her. Lastly, when she discovers that Darcy helped her family out in that situation, and that his tenants love and adore him for his kindness, she is completely humbled and realizes that she cannot be prideful any longer.
Darcy begins the novel with set prejudices: he feels that country people are uneducated and unworthy of companionships; he feels that Elizabeth's family is degrading and embarrassing; he thinks that his money should earn him a bride. He holds those prejudices for much of the novel. It isn't until he is rejected by the "poor" Elizabeth, and soundly chastised for being snobby to his sister that he realizes that he might have indeed been a bit elitist in the situation, and presumptuous about his own knowledge of other people. Her lecture strikes his heart and he learns to regret his haughty attitude. He makes up for it in helping Elizabeth's family with Lydia, and in being kind to her aunt and uncle when they visit. He shows he is truly reformed when he stands up to his regal Aunt Katherine and marries Elizabeth against her wishes.
Both characters, through a journey together, learn to be humble and less judgmental. I hope that those thoughts helped; good luck!