It felt most strange to stand here in the silence and know that he was about to leave the house for the last time. Long ago, when he had been left alone while the Dursleys went out to enjoy themselves, the hours of solitude had been a rare treat: Pausing only to sneak something tasty from the fridge, he had rushed upstairs to play on Dudley's computer, or put on the television and flicked through the channels to his heart's content. It gave him an odd, empty feeling to remember those times; it was like remembering a younger brother whom he had lost. (4.2)
Harry's thoughts on leaving Number four, Privet Drive, his longtime home-that's-not-a-home, are bittersweet; clearly, with this departure, an epoch of his life is over, and can never be reclaimed.
He had a strong, though inexplicable, feeling that [Godric's Hollow] held answers for him. Perhaps it was simply because it was there that he had survived Voldemort's Killing Curse; now that he was facing the challenge of repeating the feat, Harry was drawn to the place where it had happened, wanting to understand. (6.60)
Godric's Hollow is an intriguing idea to Harry – it's both a symbol of home and of danger, and Harry's not sure what he'll find there.
Harry's extremities seemed to have gone numb. He stood quite still, holding the miraculous paper in his nerveless fingers while inside him a kind of quiet eruption sent joy and grief thundering in equal measure through his veins. Lurching to the bed, he sat down.
He read the letter again, but could not take in any more meaning than he had done the first time, and was reduced to staring at the handwriting itself. She had made her "g's" the same way he did: he searched the letter for every one of them, and each felt like a friendly little wave glimpsed from behind a veil. The letter was an incredible treasure, proof that Lily Potter had lived, really lived, that her warm hand had once moved across this parchment, tracing into these letters, words about him, Harry, her son. (10.16)
Finding this letter from Lily is like discovering traces of a home Harry's never had – this is a tangible piece of evidence, like an extension of his mother herself.
The orphanage had been the place Voldemort had been determined to escape; he would never have hidden a part of his soul there. Dumbledore had shown Harry that Voldemort sought grandeur or mystique in his hiding places; this dismal gray corner of London was as far removed as you could imagine from Hogwarts or the Ministry or a building like Gringotts, the Wizarding bank, with its golden doors and marble floors. (15.25)
The "homes" Voldemort chooses for his Horcruxes are incredibly significant – in a way, they fit with the old saying, "home is where the heart is." Apparently, for Voldemort, the ideal home is where the Horcrux is… in places of great wizarding significance that represent the power of the magical world. I guess we never expected the guy to be truly sentimental anyway.
He was about to go home, about to return to the place where he had had a family. It was in Godric's Hollow that, but for Voldemort, he would have grown up and spent every school holiday. He could have invited friends to his house… He might even have had brothers and sisters… It would have been his mother who had made his seventeenth birthday cake. The life he had lost had hardly ever seemed so real to him as at this moment, when he knew he was about to see the place where it had been taken from him. (16.42)
The loss of Harry's real home feels even worse than ever when he's faced with the prospect of finally going to Godric's Hollow. We see here the loneliness of an orphaned boy, who's always just wanted nothing but the family he lost so long ago.
[Voldemort] walked on, around the edge of the lake, taking in the outlines of the beloved castle, his first kingdom, his birthright… (24.150)
Voldemort's insistence that Hogwarts is his "first kingdom, his birthright" demonstrates his warped attitude towards home; for him, it's a place that only he can truly inhabit.
Tom Riddle, who confided in no one and operated alone, might have been arrogant enough to assume that he, and only he, had penetrated the deepest mysteries of Hogwarts Castle. Of course, Dumbledore and Flitwick, those model pupils, had never set foot in that particular place, but he, Harry, had strayed off the beaten track in his time at school – here at last was a secret he and Voldemort knew, that Dumbledore had never discovered –. (31.87)
Again, Harry proves his odd kinship with Voldemort – both of them know the Castle's secrets better, perhaps, than anyone else, since it's home to the two of them. It's taken all of Harry's transgressions as well as his good deeds to get this close to Voldemort, and to figure out the Dark Lord's secrets.
He wanted to be stopped, to be dragged back, to be sent home…
But he was home. He and Voldemort and Snape, the abandoned boys, had all found home here. (34.31-32)
Hogwarts is home to Harry, and there's no other place for him to go. Abandoned and orphaned once, he has come back to the only place he truly knows. And, poetically, he returns to face another orphan whose only real home was this school.