Maybe they are following a lead on a new dark magic threat. If there is a new and dangerous form of dark magic emerging in America, given their track record of fighting against dark forces, they would be the ones to go and investigate. They would use their skills and intelligence to figure out what's going on and how to stop it.
They could be on a mission to exchange magical knowledge. Hogwarts has its own set of spells and magical techniques, and so does the American wizarding world. They might go to America to learn new things and share their knowledge as well.
It might be related to Hermione's research. She may have read about some unique magical phenomenon in Australia that she wants to study first - hand. And Harry, being her loyal friend, goes along with her. For example, there could be a special type of plant with healing properties that only grows in certain parts of Australia.
It could be for safety reasons. If there's a threat at Hogwarts or in the wizarding world in general, Hermione's house might be a safe haven. Her family might have some protective spells around the house that Harry can take advantage of. Also, Hermione's parents are Muggles, so it might be a place where the dark forces wouldn't expect them to hide.
Perhaps they get involved in the American wizarding politics. The political situation in the American wizarding world might be different from that in the UK. Harry and Hermione, with their experience from the fight against Voldemort, could contribute to solving some problems in the American wizarding community, like issues related to the secrecy of magic in a more urbanized and diverse American society.
One reason could be to search for a particular magical item. Maybe Dumbledore left behind a clue that led them to believe there was something important in one of the vaults at Gringotts.
Perhaps he's following a lead related to his parents. There could be some secrets about his family that are hidden in America. He goes there to uncover the truth and learn more about his family's past that might not be available in the UK.
Maybe Hermione chooses Ron over Harry in the fanfiction. Since Harry has deep feelings for Hermione, this would be a crushing blow to him.
They might go into hiding because there are still some remnants of Voldemort's followers who are seeking revenge on Harry. These followers could be more dangerous than before as they are desperate and hiding in the shadows. Ginny, being Harry's wife, is also in danger, so they choose to go into hiding to avoid any potential attacks.
One reason could be to study rare magical plants. Australia has a diverse range of flora, and some of it might have magical properties that are not found anywhere else. So, he travels there to learn about these plants and their potential uses in potions or other magical practices.
They might go to the Burrow to reconcile. Fanfiction often explores the idea of enemies becoming friends. The Burrow, being a place full of warmth and family, could be a good setting for them to start over. It's a place where they can put aside their differences and get to know each other better without the pressure of Hogwarts politics.
Perhaps he was on a quest for a rare magical ingredient only found in PT.