Miss Havisham has a problem with men. She was left at the altar when she young. Now, she's a man-hater. Having Estella as a daughter, she wants to teach Estella to wreak havoc on men. She is training Estella through these excursions with Pip to taunt, tease, mislead, mock and put-down someone of the other gender.
Pip misreads this as he gets older. As a youngster, he was never quite sure what to make of it. Pip as a child in a lower class also helps give Estella something to mock. She makes fun of how he plays cards, what he calls jacks, and what his shoes look like.
The two reasons why Miss Havisham wanted Pip to come to Satis House are:
1. In Ch.7 we read that Miss Havisham is alone and is bored and would like to be entertained, so as a means of relief for her boredom she asks Pumblechook, her tenant, to send a small boy to her house who would entertain her and relieve her boredom. Pumblechook informs Mrs. Joe who in turn sends Pip to Miss Havisham's house:
`And couldn't she [Miss Havisham] ask Uncle Pumblechook if he knew of a boy to go and play there? Isn't it just barely possible that Uncle Pumblechook may be a tenant of hers, and that he may sometimes -- we won't say quarterly or half-yearly, for that would be requiring too much of you -- but sometimes -- go there to pay his rent? And couldn't she then ask Uncle Pumblechook if he knew of a boy to go and play there? And couldn't Uncle Pumblechook, being always considerate and thoughtful for us -- though you may not think it, Joseph,' in a tone of the deepest reproach, as if he were the most callous of nephews, `then mention this boy[Pip], standing Prancing here'
`She [Miss Havisham] wants this boy [Pip] to go and play there.
2. In Ch.44 Pip is able to compel Miss Havisham to confess that she used Pip as a means to torment her avaricious and selfish relations namely, Sarah Pocket, Miss Georgiana, and Mistress Camilla:
I was liberally paid for my old attendance here,' I said, to soothe her, `in being apprenticed, and I have asked these questions only for my own information. What follows has another (and I hope more disinterested) purpose. In humouring my mistake, Miss Havisham, you punished -- practised on -- perhaps you will supply whatever term expresses your intention, without offence -- your self-seeking relations?'
`I did. Why, they would have it so! So would you.