He turned to smile at the tabby, but it had gone. Instead he was smiling at a rather severe-looking woman who was wearing square glasses exactly the shape of the markings the cat had had around its eyes. (1.47)
Rowling doesn't waste any time – we're only partway through chapter one and we've already witnessed our first Transfiguration. Here McGonagall transforms from a tabby cat into her real form – professor at Hogwarts. From context given later in the book, we know how hard just transforming a match into a needle is: given the ease with which McGonagall goes from cat to person, she must be a very powerful witch indeed.
"I'm a what?" gasped Harry.
"A wizard, o' course," said Hagrid, sitting back down on the sofa, which groaned and sank even lower, "an' a thumpin' good 'un, I'd, say, once yeh've been trained up a bit. With a mum an' dad like yours, what else would yeh be?" (4.63-64)
For Harry – and, we might imagine, anyone who's grown up only knowing about Muggles – learning that there's not only magic and wizards, but that he himself is one, comes as a pretty big shock. It's not just finding out that supernatural elements exist, it's discovering that he's one of those supernatural elements himself.
"Wizards have banks?"
"Just the one. Gringotts. Run by goblins."
Harry dropped the bit of sausage he was holding.
We're not sure what's more of a shock here: 1) the idea that "wizards have banks," just like non-wizards do, and that they would still value currency the way non-wizards do; or 2) that there are "goblins" in this world, and that they work at the wizard bank. For Hagrid, of course, who's grown up in this magical realm, it's just a matter of course. And really, for Harry, it shouldn't be too much of a stretch to start believing in goblins, when he's already been convinced about wizards.
"I don't know how the Muggles manage without magic," he said as they climbed a broken-down escalator that led up to a bustling road lined with shops. (5.76)
While Harry's still getting used to the idea that magic exists and witches and wizards are real, Hagrid reminds us that this magic is ancient and long-established, and he can't even figure out how people get along without it. This goes a long way toward establishing the tremendous size and depth of this new world, which Harry's found himself a part of.
Harry took the wand. He felt a sudden warmth in his fingers. He raised the wand above his head, brought it swishing down through the dusty air and a stream of red and gold sparks shot from the end like a firework, throwing dancing spots of light on to the walls. (5.244)
This is the first time that Harry gets a feel for doing magic himself and the potential thrill that casting charms can bring. He's seen plenty of other awesome magical things happen so far – just being in Diagon Alley is pretty cool – but this is the first kind of magic he's gotten to experience as someone who causes it, rather than someone who watches the effects.
And then, once you had managed to find them, there were the classes themselves. There was a lot more to magic, as Harry quickly found out, than waving your wand and saying a few funny words. (8.12)
Lest we think that becoming a wizard is all fun and games, Hogwarts reminds us, through Harry, that it's actually very hard work. The students at Hogwarts may have innate wizarding talent, or potential, but that doesn't translate to experience and success right out of the gate. In a way, that's kind of reassuring.
Ron also started teaching Harry wizard chess. This was exactly like Muggle chess except that the figures were alive, which made it a lot like directing troops in battle. (12.45)
This is just a small example of how the wizarding world is filled with magic, down to the board games children play. Even the small playing pieces are "alive." When you consider how a single "live" children's toy is often the inspiration for a whole book (see The Velveteen Rabbit, The Indian in the Cupboard, The Return of the Twelves, etc.), it's pretty amazing that the Harry Potter series is so rich and full of detail that this abundant topic becomes another small detail in the magic tapestry Rowling paints.
'I don't need a cloak to become invisible,' said Dumbledore gently. 'Now, can you think what the Mirror of Erised shows us all?' (12.189)
In case we needed any reminding about just how powerful Dumbledore is, he tells Harry that he doesn't "need a cloak to become invisible." Harry depends on an object with its own magical properties – he or anyone else could use it with the same effects. Not so for Dumbledore's invisibility: this is magic the first-year students can't even dream of.
The ancient study of alchemy is concerned with making the Sorcerer's Stone, a legendary substance with astonishing powers. The stone will transform any metal into pure gold. It also produces the Elixir of Life, which will make the drinker immortal. (13.50)
This quotation from a book Hermione gets from the Hogwarts library doesn't say anything that different from what a Muggle book about alchemy might say. The philosophical study of alchemy, after all, can also be thought of as a science. The difference is that, in the magic world, phrases like "will transform… into gold" and "will make the drinker immortal" aren't just "legendary"; it really works.
It was one of my more brilliant ideas, and between you and me, that's saying something. You see, only one who wanted to find the Stone – find it, but not use it – would be able to get it, otherwise, they'd just see themselves making gold or drinking Elixir of Life. My brain surprises even me sometimes… (17.137)
Here, Dumbledore inserts a little humility into the proceedings, as he pokes fun of the fact that he has so many "brilliant ideas" and lets Harry into the little secret that this one was one of the best. We're reminded, once again, of his power and wisdom as both a great wizard and as Headmaster of Hogwarts. Of course, his plan backfired a little bit, as Harry found himself holding the Stone in Voldemort's presence, but otherwise there was no harm done – at least, not this time, not to the good guys.