He could have an amazing first year. He might be sorted into Gryffindor like his parents. And then he quickly makes friends with some interesting characters. He could also be really good at a particular subject like Potions, just like Snape was.
It could make his first year more exciting. Tonks might give him some advice on how to deal with Snape, which would be really helpful for Harry.
Their relationship with the Hogwarts student body as a whole would be a big part of the story. Some students might gossip and spread rumors, which could isolate Harry and Hermione. But there would also be those who would be empathetic. Their relationship with each other would also deepen in new ways. They would have to rely on each other more than ever, and the shared experience of pregnancy would bring them closer together, even as they face the challenges of being young parents - to - be in a magical school.
In this fanfiction, Harry's life at Hogwarts would be full of new discoveries. He could find secret passages that are overgrown with plants but are only visible to him because of his connection to Demeter. His friends might be a bit in awe of his new powers. He'd also have to deal with some jealousy from other students who don't have such unique abilities. And when it comes to Quidditch, he could use his nature powers to his advantage, like creating a gentle breeze to help him fly faster or making the grass on the pitch grow in a way that confounds his opponents.
There are various reasons in fanfiction for Hermione to be jealous of Harry's girlfriend. One reason could be that she's jealous of the easy, romantic relationship Harry has with his girlfriend, while her own love life might be a mess. Also, she might be afraid that the girlfriend will come between their friendship. The fanfiction could show her acting out in small ways, like being a bit snippy with Harry or trying to find flaws in his girlfriend.
Hermione would likely be very organized. She'd have a schedule for cleaning and use specific spells for different tasks. Harry might be more haphazard but still try to do a good job because of Hermione's influence.
He might face challenges similar to what Harry did. Maybe he has to deal with a bully who mocks his parents' fame. Just like Harry had Draco, his son could have someone who gives him a hard time right from the start.
They would raise him with strong moral values. Given their own experiences, they would teach him about the importance of standing up against evil and the value of friendship. Harry would likely pass on his Quidditch skills and love for the sport, while Hermione would ensure he has a great education in magic.
Hermione's academic performance might improve even more. Sirius could provide her with rare books and knowledge from his family's library. For example, there could be ancient spells and magical theories that are not commonly known. This would give her an edge in her classes.
In some versions of this fanfiction, it could be Neville. He also had a difficult family situation, so he might be more attuned to signs of distress in Harry. Maybe he notices Harry's worn - out clothes that are too big for him, which could be a sign of neglect, and then starts to piece together that something is wrong at the Dursleys'.
In post - Hogwarts fanfiction, Harry often develops into a more confident leader. He has already defeated Voldemort, so he uses his experience to mentor the next generation. Hermione, on the other hand, may become more assertive in the academic and political aspects of the wizarding world.