She could run into students from different magical families with their own unique traditions. These students could expose Hermione to new ways of thinking about magic and its place in the wizarding world, expanding her understanding of the magical community as a whole.
Perhaps she'll meet older, more experienced students who can act as mentors. These students could introduce her to secrets of the new school that are not easily accessible to newcomers. They might also help her with any homesickness she has for Hogwarts.
She could meet Bella Swan. They might start off as a bit of an odd couple, with Bella's more reserved nature compared to Hermione's boldness. But they could learn a lot from each other.
She could meet the headmistress of St Trinians. That would be an interesting encounter as Hermione tries to understand the school's rules and the headmistress tries to figure out this strange, book - smart girl. Also, she might meet the troublemakers of the school, those girls who are always getting into scrapes and might see Hermione as either a threat or someone to rope into their schemes.
Well, she might face new challenges. In a different school, the curriculum could be completely different. Maybe she'll have to learn new spells or magical theories that are unique to that institution. She could also meet new friends or rivals. There might be a different social hierarchy compared to Hogwarts, and she'll have to find her place in it.
Hermione might also come across some magical creatures that are guarding certain vaults in Gringotts. For example, a dragon that she has to find a way to bypass without getting burned. These encounters would add more excitement and danger to her visit to the wizarding bank.
There's a chance she meets some old Death Eaters who are lurking around Spinners End. This would put her in a very dangerous situation, but it would also make for an exciting plot. She would have to use her wits and magic skills to outsmart them. Or perhaps she meets some of Snape's old school friends who can provide insights into his character and motives.
One of her friends could be a girl who is also interested in Muggle Studies, which is not really a popular subject in Durmstrang. They meet in the library while looking for books on the topic and start sharing their views on the Muggle world and magic. Also, there might be a young wizard who is against the more aggressive and dark - leaning teachings of Durmstrang and sees Hermione as an ally in changing the school's culture.
Well, perhaps the plot could involve Hermione facing challenges in a male - dominated military school environment. She might use her intelligence to outwit bullies and excel in military tactics and training.
Madame Maxime, the headmistress of Beauxbatons, would surely play an important role. She might be initially skeptical of Hermione but later come to respect her. Also, there could be a young male wizard at Beauxbatons who is intrigued by Hermione's intelligence and Hogwarts - style magic. He could try to learn from her and in the process, they become friends and face challenges together in the school.
Obviously Hermione herself. There could also be the military school instructors, who might be strict and tough. And some of her classmates, perhaps some are friendly and others are rivals.
She could interact with the founders of Hogwarts. Imagine Hermione having a conversation with Rowena Ravenclaw about knowledge and intelligence. It would be a really cool scene as they could exchange ideas about magic and education.