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.
One possibility could be that Hermione uses her magic skills in Forks in unexpected ways. Maybe she helps the local people solve some strange problems that they can't figure out with normal means. For example, she could use a spell to find a lost item in the forest which is full of mystery in Forks.
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.
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.
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.
Well, one possible plot could be Hermione using her magic skills to solve the mysteries in Forks. Maybe she senses there's something strange about the Cullens and uses her knowledge to figure out they're vampires. She could also use her intelligence to outwit the high school bullies or help her new friends with their academic problems.
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.
It could be some traditionalist wizards who don't like the idea of Hermione and Teddy leaving the wizarding world and living among muggles. They might try to bring them back by force or create problems for them in Forks. Another possibility is some local dark forces in Forks that are threatened by Hermione's magic and Teddy's abilities.
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.
At first, Hermione would likely be a bit of an outcast. But she'd soon start to make an impression. She'd adapt by using her study skills to help other students. In the library, she'd be a regular, just like at Hogwarts. She might even introduce some new study methods to the school. And as she gets to know the students better, she'd start to understand their worldviews and adapt her own behavior accordingly, while still maintaining her core values from the wizarding world.
Voldemort's remaining Death Eaters could be behind it. They might see Hermione as a threat to their attempts to regain power in the wizarding world.