Perhaps Snape would test Harry first. He's not one to easily offer help. He might make Harry prove his worthiness by performing a difficult potion - making task or answering some really tough questions about magic theory. If Harry passes, Snape would grudgingly start to assist him, but still with his usual sarcastic remarks.
In the fanfiction, Snape could be in a really tough situation when Harry comes to him for help. On one hand, he has his loyalty to the Dark Lord (or at least that's what others think), and on the other, there's this boy who is the son of the woman he loved. He might end up guiding Harry in a very roundabout way, like giving him a potion ingredient that has a hidden meaning or leading him to a certain book in the library that contains the solution to Harry's problem.
He might encounter some strange magical creatures that are only found underground. For example, there could be a new type of goblin or a magical mole with special powers.
He could encounter a new magical creature. The goblins at Gringotts are known to have various magical creatures guarding different vaults. Maybe Harry comes across a rare and friendly creature that helps him on his journey. For example, a small fire - breathing dragon that he befriends and it later aids him in a battle against Voldemort's forces.
Well, perhaps Snape would be even more surly during the summer. He might assign Harry extra potions work as a form of punishment or to keep him out of trouble. Harry, on the other hand, would probably be more rebellious in a fanfic scenario during summer, as he's away from the strictures of Hogwarts. Maybe he'd try to uncover some secrets about Snape that he suspects.
He might meet some strange magical creatures there that are used for experimental treatments.
In some fanfictions, when Harry Potter goes through the mirror, he might find himself in a parallel magical world. This world could have different versions of his friends and enemies, and he may have to face new challenges to find his way back to his own world.
Well, Harry could be really confused at first. Ouran is a very different world from Hogwarts. He might be shocked by the opulence of Ouran Academy and the strange club system there. Maybe he'd try to find a way to fit in while still being true to his wizard self.
In fanfic, when Harry Potter goes to Gringotts, he might discover some long - lost secrets about his family's vault. There could be hidden spells or artifacts that were left there by his parents or ancestors.
Maybe Harry has some strange magical illness that normal wizards can't cure. So he has to seek help from a very special doctor who might have some unique remedies.
Perhaps he would be seen as an oddity at Alfea. After all, he comes from a completely different magical system. But his reputation as the 'Boy Who Lived' might precede him. Some might be in awe of him, while others might be skeptical. His presence could also stir up some rivalries among the students at Alfea who are used to their own hierarchies. Harry would have to find his place among the fairies and deal with any potential jealousies or hostilities.
Well, he might encounter new magical creatures unique to America, like the Thunderbird. He could also visit American wizarding schools and be amazed by their different teaching methods compared to Hogwarts.