After a few days, Harry adjusts to his freedom. He spends his days exploring the shops in Diagon Alley. He does his homework at Florean Fortescue's Ice Cream Parlor, and Fortescue himself helps Harry with his essay on witch hunts. He has to exercise control to not purchase the newest racing broom, the Firebolt, which is on display in the Quidditch shop. Harry does buy new robes, potions supplies, and schoolbooks. He's surprised to find The Monster Book of Monsters at the bookshop, but then realizes that it's the assigned text for Care of Magical Creatures.
Now that Harry knows The Monster Book of Monsters is an assigned textbook, Hagrid's cryptic note makes much more sense. This illustrates on a very small scale how simply having another shred of information can be enough to entirely change a person's perspective on something. The mention of witch hunts again reminds the reader that Harry will learn this year about how ridiculous "justice" can be.
The manager of the shop is relieved when Harry says he already has The Monster Book of Monsters. As the manager gets Harry his textbook for Divination, Harry notices the cover of another book and stares in shock: the book is about death omens, and the cover picture is of a hulking black dog. Harry rushes back to the Leaky Cauldron with his books, dumps them on his bed, and then looks in the mirror. He tells himself that he just saw a stray dog in Magnolia Crescent.
Again, now that Harry knows that a big black dog is a death omen, he's able to make more sense of past events and, more importantly, can look forward to the future and understand better what's going on if he does see the black dog again.
Harry begins looking for Ron and Hermione as the start of term approaches. He finds Seamus Finnigan and Dean Thomas, as well as Neville Longbottom. Finally, on the last day of the holiday, he runs into Ron and Hermione at the ice cream shop. Hermione immediately asks about Aunt Marge while Ron howls with laughter. Ron doesn't know why Fudge didn't punish Harry for his transgression—Mr. Weasley works at the ministry and so might have insight. Ron does say that the Ministry is providing cars to drive them to the train station tomorrow and invites Harry to join them.
Though Ron doesn't know about Black’s connection with Harry, it is likely that the adults in Ron's life do know. This reminds the reader that, because Harry, Ron, and Hermione are still children, what they learn about the world is still heavily filtered through their parents and other adult authority figures. The cars to get to the train station suggest that security is higher than it would normally be, though Ron's inability to notice this points to his immaturity.
Ron pulls out a box containing his new wand and Harry asks about Hermione's bulging bag of books. She begins listing all the classes she's taking and Harry asks if she's planning on eating or sleeping. Changing the subject, Hermione says she'd like to buy an owl, since her parents gave her money as an early birthday present. They head to the magical creature shop. Ron takes Scabbers, who is lethargic and droopy, to the counter so the witch can look him over. The witch asks how old Scabbers is and if he has any powers. Scabbers is twelve and, in the years that the Weasleys have had him, has shown no powers.
Note that Harry is aware that Hermione's course load seems ridiculous. This indicates that he's able to observe things that are off, but he's not necessarily able to draw any conclusions about what he notices (he doesn't learn until the end of the novel that Hermione will use time travel to make it to all her classes).
The witch says that a garden rat like Scabbers will only live for about three years and offers Ron some rat tonic. Before Ron can accept it, a huge orange cat leaps onto his head and then makes a dive for Scabbers. Scabbers races out of the shop, Harry and Ron in hot pursuit. Once they catch the rat, they head back to the shop. Hermione comes out as they get close but instead of an owl, she has the orange cat. Glowing, she calls the cat Crookshanks and gives Ron his rat tonic, insisting that Scabbers will be fine.
Notice that, according to this witch, Scabbers has already outlived all expectations. When neither Ron nor Harry see anything strange about Scabbers's unusually long lifespan, it indicates that when something (like Scabbers's presence) is normalized, it becomes more difficult to ask questions about why exactly it is considered normal.
Harry, Ron, and Hermione find Mr. Weasley in the bar at the Leaky Cauldron. Harry asks about Sirius Black when he notices a photo in the paper again. Mr. Weasley sighs that all Ministry employees are on the lookout and shuts Ron down when he asks about a reward for catching him. Mrs. Weasley, Fred, George, Percy, and Ginny come into the bar, all carrying shopping bags. Percy, who has been made Head Boy, greets Harry pompously, and Fred and George mimic him. Mrs. Weasley says with pride that Percy is the second Head Boy in the family. When she points out that Fred and George weren't made prefects, they say that'd take all the fun out of life.
Mr. Weasley's quick move to shut down Ron's idea about catching Sirius Black himself shows that he's well aware that Harry and Ron already have a reputation of getting into trouble when there's trouble to be had. This tells the reader that all the adults in Harry and Ron's life are doing their best to keep them safe, as they know that things could quickly get dangerous if they're allowed to ask too many questions.
For dinner that night, the entire Weasley family eats with Harry and Hermione. They discuss transportation to King's Cross Station and Mrs. Weasley sends Ron to pack. As Harry gets to his room, he hears Ron and Percy fighting next door. Harry goes to investigate and learns that Percy's Head Boy badge and Scabbers's rat tonic are gone. Harry offers to go look in the bar for the rat tonic but as he passes a parlor, he hears Mrs. and Mr. Weasley fighting too.
Again, notice that Harry doesn't question why the Ministry is escorting the Weasleys to the train station like this—and most importantly, it doesn't occur to Harry that the escort is actually for his benefit, not the Weasleys'. However, Harry doesn't make this leap because he trusts the adults around him and feels safe.
Harry hesitates to listen until he hears Mr. Weasley say that Harry has a right to know. Mrs. Weasley doesn't agree, but Mr. Weasley says it's for Harry and Ron's own safety. He points out that Sirius Black is mad and before he escaped, he was muttering "he's at Hogwarts" in his sleep—Black is after Harry and seems to believe that killing Harry will bring back Voldemort. Mrs. Weasley points out that Harry will be safe while Dumbledore is around and Mr. Weasley concedes that the Azkaban guards will be at Hogwarts, thereby making Harry safer. Harry hurries to find the rat tonic.
This conversation shows Harry that his trust in the adults around him and in the Ministry has blinded him to what's actually going on. Though it's likely that everyone is just trying to keep Harry safe, it's worth noting that now that Harry has this information, he'll be able to make more informed choices about what he does, where he goes, and how seriously he takes adults' attempts to protect him.
When Harry gets back upstairs, he finds Fred and George on the landing with Percy's badge—it now reads "Bighead Boy." He forces a laugh, gives Ron the rat tonic, and then shuts himself in his room. Everything makes sense now—Fudge was so lenient because Black is after Harry. Harry wonders why he doesn't feel more scared, but he does agree with Mrs. Weasley that with Dumbledore, he'll be safe. He also wonders what the deal is with the Azkaban guards, since everyone seems terrified of them. What bothers Harry most is that it seems likely that nobody will want him to visit Hogsmeade. He scowls, thinking that he's already escaped Voldemort three times, and flashes on the big dog he saw in Magnolia Crescent. He says out loud that he's not going to be murdered.
Harry's thought that he's already escaped Voldemort three times indicates that though he's a child, he does have an inflated sense of his own abilities and importance—it implies that he thinks the adults are being silly and that Harry doesn't see himself as needing protection. This suggests that as Harry grows, one of the things he'll have to learn is how to both ask for and accept help in situations like this as he can't do things entirely alone.