Well, they might be shocked at how much technology has advanced in the future depicted in the fanfiction. Maybe there are new types of elemental powers or different ways of training ninjas. It could also be that the relationships between characters have changed in unexpected ways, and they would have to come to terms with that.
They might gain a new perspective on their own abilities. If the future fanfiction shows ninjas with enhanced powers, the past characters could start to think about how they could develop their skills further. For example, if in the future they can combine elemental powers in new ways, the past ninjas might start experimenting.
He might start by observing. Aang is a curious person, so he'd likely watch how people live, how they interact with technology. He may also try to find other benders to learn about any new developments in bending. And he'd probably use his kind heart to make friends easily in this new future environment.
One idea could be a crossover where Ben 10's Omnitrix malfunctions near Avatar Aang's world. Ben and Aang team up to face a new threat that is causing chaos in both their realms. Maybe some of Ben's alien forms have unexpected interactions with the bending abilities in the Avatar world.
In a fanfiction, the Avatar State of Aang could be used in a more creative way. For example, Aang accidentally enters the Avatar State during a cultural festival. His friends are initially worried, but then they realize that in this state, Aang can communicate with the spirits present at the festival in a unique way. He learns secrets about the spirits and their connection to the human world, which he then uses to strengthen the bond between the two realms when he's out of the Avatar State.
One great Aang fanfic could be one where he has to face an unexpected new threat in the Spirit World. It might explore his growth as the Avatar even further, showing his unique connection to the spirits and how he uses airbending in creative ways to overcome challenges.
Yes, there are quite a few interesting 'Avatar Aang and Katara' fanfictions out there. You can start by looking on popular fanfiction websites like Archive of Our Own. Some fanfictions explore their relationship further after the events of the show, delving into their daily lives and how they grow together.
Another fanfiction could explore the challenges they face during Katara's pregnancy. Since Aang is the Avatar, he has many responsibilities. There could be a conflict where he has to leave for an important mission, but he doesn't want to leave Katara alone. Katara, being strong - willed, tells him to go and fulfill his duties. In his absence, she discovers new water - bending techniques related to pregnancy and childbirth, which she plans to teach other women in their community later.
He could meet his future self. It would be an interesting encounter as he could learn from his future experiences and wisdom. Or he might meet people who have studied his past adventures and see himself through their eyes.
One common theme is power control. Aang often has to learn how to handle the immense power of the Avatar State without losing himself. For example, in many fanfictions, he struggles with the sudden influx of energy and the potential for destruction it holds.