"It's the best place for him," said Dumbledore firmly. "His aunt and uncle will be able to explain everything to him when he's older. I've written them a letter." (1.81-82)
It's almost touching that Dumbledore would be so naive as to think that a "letter" from him would smooth things over with the Dursleys and make everything OK. As we find out by chapter two, the last thing their house feels like, to Harry, is a home. Minor spoiler: later on in the series, we learn that Dumbledore had another important reason why Harry had to spend time with the Dursleys and why it really was "the best place." But that's all we're going to say for now.
The room held no sign that another boy lived in the house, too. (2.1)
Yuck. The Dursleys are just icky. They've filled their house with pictures of and references to their favorite son Dudley, but they won't even let Harry have a single picture of his parents. The Dursleys don't make their home welcoming to their poor orphaned nephew in any way.
Harry was used to spiders, because the cupboard under the stairs was full of them, and that was where he slept.
When he was dressed he went down the hall into the kitchen. The table was almost hidden beneath all Dudley's birthday presents. (2.13-14)
These two paragraphs show how different Harry and Dudley's positions in the house are. Harry doesn't even have a bedroom. He sleeps in a "cupboard" "under the stairs" – we can only imagine how dark and dismal it is. Think of how many spiders he must have encountered in order to get "used to" them. In contrast, Dudley gets so much attention that there's barely enough room in the house for all his birthday presents. (Compare this birthday to Harry's pivotal eleventh non-celebration, too.)
Harry spent as much time as possible out of the house, wandering around and thinking about the end of the holidays, where he could see a tiny ray of hope. When September came he would be going off to secondary school and, for the first time in his life, he wouldn't be with Dudley.
Dudley had been accepted at Uncle Vernon's old private school, Smeltings. (3.3)
We can't even begin to think about how dreadful it would be to go to Smeltings with Dudley. Luckily, Harry doesn't have to. This passage reveals that Harry was excited about going to school even before he knew of Hogwarts – just the thought of getting away from Dudley gives him "a tiny ray of hope." If just going to a different school is exciting, how much more awesome is going to wizarding school going to be?
…Harry could hardly believe it when he realized that he'd already been at Hogwarts two months. The castle felt more like home than Privet Drive ever had. (10.66)
Sixty days at Hogwarts, and Harry already feels more comfortable there than he did in eleven years at his aunt and uncle's. That says a lot about how unhappy he was at the Dursleys'; after such a short while, Hogwarts is already home. It's hard to say if other students also feel as at home at Hogwarts as Harry does, but Harry's definitely not alone in his excitement to arrive and reluctance to leave.
"I do feel so sorry," said Draco Malfoy, one Potions class, "for all those people who have to stay at Hogwarts for Christmas because they're not wanted at home." …
It was true that Harry wasn't going back to Privet Drive for Christmas… He didn't feel sorry for himself at all; this would probably be the best Christmas he'd ever had. (12.3, 12.5)
Malfoy is trying to insult Harry and the other students who are staying at Hogwarts over the break, but once again he falls short of the mark. Harry may not be "wanted" at the Dursleys, but since he doesn't consider it home, it's not much of an insult. He doesn't want to go there either. In fact, he doesn't even refer to his aunt and uncle's house as a home, calling it by its street name ("Privet Drive") instead of a home or a house. While Malfoy claims to "feel so sorry" for everyone who's stuck at Hogwarts, Harry is excited for what he thinks will be the "best Christmas" of his life.
He looked very pleased with himself, but Hermione didn't.
"Hagrid, you live in a wooden house," she said. (14.51-52)
Here, Hermione reminds us of the literal importance of kinds of homes. Hagrid's home is unprepared for dragon raising for many reasons, not least of all the issue that the dragon breathes fire constantly, and the house is made of wood. His home has structural problems with the very idea of dragon raising. His idealism shows us both his big heart and total lack of practicality.
A hundred and fifty points lost. That put Gryffindor in last place. In one night, they'd ruined any chance Gryffindor had had for the house cup. Harry felt as though the bottom had dropped out of his stomach. How could they ever make up for this? (15.16)
Although Hogwarts serves as a big home for all the students during the academic year, students also feel a responsibility to their individual "houses." Each of the four Hogwarts houses is a kind of home containing a kind of family (true for at least the academic year). Here, Harry was trying to do something good, but instead has let down both his school home and school family terribly.
It seemed to be a handsome, leather-covered book. Harry opened it curiously. It was full of wizard photographs. Smiling and waving at him from every page were his mother and father.
"Sent owls off ter all yer parents' old school friends, askin' for photos… knew yeh didn' have any… d'yeh like it?"
Harry couldn't speak, but Hagrid understood. (17.175-177)
We're going to go unconventional here, so bear with us. The text has already established that Harry doesn't feel at home at the Dursleys'. While he does feel at home at Hogwarts, he also knows he's at school, and that school has a fixed term. In other words, Hogwarts isn't his forever home. Since Harry isn't fully established in a permanent, physical residence, he's open to thinking of other kinds of homes. With this kind gift from Hagrid, Harry has pictures of his parents for the first time, and in this album, he always has a "place" where he can come home.
Harry knew at once that Snape's feelings toward him hadn't changed one jot. This didn't worry Harry. It seemed as though life would be back to normal next year, or as normal as it ever was at Hogwarts. (17.201)
Having the fantasy of a perfect home is just that: a fantasy. It's too late to have the picture-perfect home and family that Harry should have started out with. So what he clings to here is the idea of what's "normal." It might seem like attending wizarding school is pretty far from normal, but it's become normal for Harry – at least, the Hogwarts definition of normal has become so for him.