Maybe they are following a lead. There could be a connection between Voldemort or other dark wizards and something in Australia. It could be a base for dark magic operations that they need to investigate. So, they travel there to put an end to it and protect the wizarding world.
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.
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 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.
One reason is that it allows for a lot of creativity. Writers can explore different aspects of the wizarding world that haven't been fully covered in the original books. For example, they can create new magical traditions or spells based on Australian Aboriginal culture. Also, the idea of Harry and Hermione in a far - away and exotic place like Australia is appealing to fans who want to see their favorite characters in new situations.
In some fanfictions, they might go there to search for something. Maybe they are looking for a magical artifact hidden in Australia. It could be a powerful charm or a lost spellbook that could help them in their fight against the Dark forces.
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.
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.
Maybe Hermione chooses Ron over Harry in the fanfiction. Since Harry has deep feelings for Hermione, this would be a crushing blow to him.
One popular 'Harry and Hermione Australia fanfiction' might be those that explore their adventures in the Australian wizarding community. Maybe they go there to study some unique magical creatures or spells that are only found in Australia. Another could be about them solving a mystery that has its roots in Australia's magical history.
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.
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.