Harry wakes up very early the next morning, dresses quickly, and goes to the kitchen. Mrs. Weasley tries to give Harry breakfast, but Harry is nervous and can barely eat. The adults discuss Order business and try to assure Harry that things will be okay. The woman conducting Harry's hearing, Amelia Bones, is fair, according to Mr. Weasley. Everyone wishes Harry luck as he and Mr. Weasley leave the house. They take the Underground into the middle of London so they can use the visitor's entrance, which is a phone box. Mr. Weasley dials a number and announces who he is. A machine deposits a badge saying that Harry’s at the Ministry for a disciplinary hearing, and he pins it to his shirt.
Recall that in previous novels, Harry hasn't been given good reason to trust the Wizarding world's justice system. Mr. Weasley is likely being optimistic to ease Harry’s nerves, then, especially given that he now knows that Sirius is innocent and of the Ministry's vendetta against Harry. It's likely that Harry’s hearing is a tool to try to discredit him, and in the process will make him relive his trauma yet again.
The box begins to lower into the ground and Harry's mouth falls open at what he sees. The atrium is long, splendid, and lined with fireplaces for employees traveling by Floo powder. There's a fountain with gold statues of a witch, wizard, centaur, goblin, and house-elf in the middle. Harry reads the sign that says money in the Fountain of Magical Brethren goes to St. Mungo's Hospital, and Harry vows to put money in if he doesn't get expelled. A security wizard checks Harry's wand and then Harry follows Mr. Weasley into an elevator.
Notice how at this point, when Harry is extremely concerned with himself and his future, he doesn't question the implications of the fountain at all (Harry will note later that it seems questionable and idealistic at best). This illustrates how a person must be able to look beyond themselves in order to identify the problems with one's society, which the fountain represents.
Mr. Weasley stops in at the Aurors' office to speak to Kingsley. Harry is shocked that the men speak to each other as if they barely know each other, but Kingsley gives Harry a wink as he gives Mr. Weasley a magazine for Sirius. Harry and Mr. Weasley head for Mr. Weasley's office. Mr. Weasley opens a memo about regurgitating toilets and explains that anti-Muggle pranksters are responsible. Perkins, Mr. Weasley's coworker, arrives and says that they changed Harry's hearing time. It's now in Courtroom Ten and started five minutes ago.
Changing Harry's hearing time and not notifying him makes it very clear that the Wizarding justice system isn't in this to serve Harry and give him a fair trial. At the very least, being late to his hearing will put Harry in a position where he looks bad and has to defend himself and his ability to act responsibly from the start. Mr. Weasley and Kingsley must act like they don’t know each other, as they’re essentially undermining the Ministry together at this point.
Mr. Weasley leads Harry away at a run back to the elevators. Mr. Weasley greets a man named Bode in the elevator and they all get off at the Department of Mysteries. Harry and Mr. Weasley take the stairs down to the lowest floor, find the courtroom, and Mr. Weasley shoves Harry inside.
The Department of Mysteries will become important later in the novel. Learning that it exists at all and where it exists at this point is an early clue that Harry at first doesn’t recognize as useful information.