The remaining Death Eaters could be the antagonists. Even though Voldemort is gone, there could be some who still hold a grudge against Harry and want to finish what their master started. They might track Harry and Ginny down as they run away, creating dangerous situations for the couple.
One possible plot could be that they run away to escape some dark magic threat that Voldemort's remaining followers are planning. They might go into hiding in a remote wizarding village, where they have to learn to fend for themselves without the help of the Order of the Phoenix. Along the way, they could discover new magical abilities or ancient spells that help them in their fight against the dark forces.
They might choose to run away secretly because they want to be alone together. They are in love, and the hustle and bustle of Hogwarts life doesn't give them enough privacy. So, they run away to a quiet place where they can enjoy each other's company without any interruptions.
Well, one possible plot could be that they run away to escape some sort of dark threat at Hogwarts. Maybe a new dark wizard has emerged and they feel that the school is no longer safe. So, they secretly gather some supplies, use magic to disguise their departure, and head off to a remote location like a hidden wizarding village in the mountains. There they try to find a way to fight the threat from afar while also exploring their relationship in a more private setting.
One reason could be to escape the public spotlight. After the war, Harry is a hero and Ginny is his girlfriend. They might feel overwhelmed by all the attention and want a quiet life, so they run away.
One reason could be to escape the constant media attention and pressure that comes with being in the wizarding world's spotlight. Harry especially has always had the eyes of the public on him, and running away might be a way to get some peace.
They could use disillusionment charms constantly to blend in with their surroundings. This would make it difficult for Voldemort's forces to spot them. Also, they might use the Marauder's Map if they manage to get hold of it. It would help them avoid any approaching Death Eaters or other threats.
One possibility is that they'd try to find other resistance fighters. They could use their skills, like Harry's Patronus, to send out signals to those who oppose Voldemort. However, they'd face many challenges. Their resources would be limited. They might have to rely on the kindness of strangers or their own wits to get food and shelter. And they'd always be in danger of being betrayed.
They might run away because they want to explore the wizarding world on their own terms. Without the restrictions of Hogwarts or the watchful eyes of the Ministry of Magic.
Ginny would likely be protective. She loves Harry and knows about his difficult past with the Dursleys. So she would probably stand up for him right away, not tolerating any of their mean remarks.