This separation from his spellbooks had been a real problem for Harry, because his teachers at Hogwarts had given him a lot of holiday homework. (1.7)
Little details like this highlight not only how awful the Dursleys are, but also how divided Harry is – he essentially exists in two worlds. We'd wager that if he attended a Muggle school, his aunt and uncle wouldn't go so far as to lock up his books for the summer.
"What are you taking Muggle Studies for?" said Ron, rolling his eyes at Harry. "You're Muggle-born! Your mum and dad are Muggles! You already know all about Muggles!"
"But it'll be fascinating to study them from the wizarding point of view," said Hermione earnestly. (4.2.19-20)
Hermione's philosophy on education seems to be to learn as much as you possibly can about every subject. And that even applies to subjects you already know a lot about – there's always more to learn. Hermione's emphasis on the importance of "point of view" here is thematically significant too (see the time travel adventure, which is all about gaining new perspectives on events).
"But everyone knows that," said Hermione in a loud whisper. Professor Trelawney stared at her.
"Well, they do," said Hermione. "Everybody knows about Harry and You-Know-Who."
Harry and Ron stared at her with a mixture of amazement and admiration. They had never heard Hermione speak to a teacher like that before. (6.1.93-5)
We really love the hardcore, rule-breaking Hermione that comes out to play during this book. Her reaction to Trelawney shows us that while Hermione respects authority (certainly more than her BFFs do) she doesn't respect it blindly. Some "authority" figures, like Trelawney, clearly aren't deemed worthy of respect.
Professor McGonagall broke off, and they saw that her nostrils had gone white. She went on, more calmly. "Divination is one of the most imprecise branches of magic. I shall not conceal from you that I have very little patience with it. True Seers are very rare, and Professor Trelawney –" (6.1.126)
Listening to McGonagall here is like listening to Hermione in a few decades, don't you think? It's fascinating how similar the two women are, in some respects. Their mutual dislike of Trelawney certainly binds them together, but it's their shared reasons for disliking Trelawney that really make the connection interesting. Both of them find Trelawney illogical and ridiculous, implying that these two gals value rationality and logic above all else.
"Why don't you just drop a couple of subjects?" Harry asked, watching her lifting books as she searched for her rune dictionary.
"I couldn't do that!" said Hermione, looking scandalized. (12.5.63-4)
The use of the word "scandalized" really sums up Hermione's over-achiever tendencies. She takes her schooling as a point of pride and quitting a subject would be like some sort of defeat to her.
Trying to answer a question with Hermione next to him, bobbing up and down on the balls of her feet with her hand in the air was very off-putting, but Harry had a go. (7.2.34)
The image of Hermione practically jumping up and down in eagerness is really vivid, as well as funny.
Rude and unmanageable as he almost always was, Peeves usually showed some respect toward the teachers. Everyone looked quickly at Professor Lupin to see how he would take it; to their surprise, he was still smiling. "I'd take the gum out of the keyhole if I were you, Peeves," he said pleasantly. "Mr. Filch won't be able to get to his brooms." (7.2.8-9)
Lupin's encounter with Peeves tells us a lot about his character, oddly enough. First, we get a sense of how cool, calm, and collected Lupin is. Second, we get a peek at Lupin's very dry sense of humor. And third, we get a point re-emphasized to us: that Lupin is pretty young and that he experienced Hogwarts before as a student. He's in a rather odd position at the moment, as an ex-student now returning as a teacher.
Harry was also growing to dread the hours he spent in Professor Trelawney's stifling tower room, deciphering lopsided shapes and symbols, trying to ignore the way Professor Trelawney's enormous eyes filled with tears every time she looked at him. He couldn't like her, even though she was treated with respect bordering on reverence by many of the class. (8.1.5)
The diction used to describe Trelawney's class, such as "stifling" and "deciphering," emphasizes just how confusing and ridiculous her class is.
Nobody really liked Care of Magical Creatures, which, after the action-packed first class, had become extremely dull. Hagrid seemed to have lost his confidence. (8.1.6)
We get some great insight into teaching here: it's largely about confidence, like so many things are in life.
They sat and made notes on werewolves from the textbook, while Snape prowled up and down the rows of desks, examining the work they had been doing with Professor Lupin.
"Very poorly explained [...] That is incorrect, the kappa is more commonly found in Mongolia [...] Professor Lupin gave this eight out of ten? I wouldn't have given it a three [...]" (9.4.36-7)
Snape once again wins Worst Teacher of the Year (and possibly of All Time) here. Though we have to admit: his running commentary on the students' essays is hilarious, albeit extremely mean.
"You're expecting too much of yourself," said Professor Lupin sternly in their fourth week of practice. "For a thirteen-year-old wizard, even an indistinct Patronus is a huge achievement. You aren't passing out anymore are you?" (12.5.4)
Contrast the cruel and bitter Professor Snape with the compassionate and encouraging Lupin, who teaches actual life lessons to Harry and his friends. Lupin acts as a true mentor to Harry and their scenes together shed light on nearly all of the book's major themes.