I liked 'Katara and Zuko's New Beginnings'. It's a fanfic that takes place after the end of the series. It focuses on how they navigate the new world order together. There are some really sweet interactions between them as they try to build a new life for themselves and their friends. It also has some great action scenes where they use their bending abilities side by side.
Zuko was the banished prince of the Fire Nation. His father, Fire Lord Ozai, burned his face during an Agni Kai when Zuko refused to fight. He was then sent on a seemingly impossible quest to capture the Avatar to regain his honor.
The setting and plot should tie in well with the Avatar universe. A fanfic that has them travel to different parts of the world, like exploring ancient Fire Nation ruins or hidden Water Tribe enclaves, can be really engaging. And of course, the dialogue has to be on point. Their conversations should be filled with the tension that comes from their different backgrounds and the new - found respect and friendship that develops over time.
You can start by looking on Archive of Our Own (AO3). It has a huge collection of fanfictions, and there are many dedicated to Zuko from 'Avatar The Last Airbender'. Just use the search function and type in 'Zuko' and the relevant tags.
Aang would feel a sense of urgency and his connection to Katara would make him even more determined. He would call upon the spirits if necessary to find her. Sokka would be full of fury and his strategic mind would be working overtime to come up with a plan. He would also be very protective of his sister's honor during this time. Toph would be steadfast in her support for the rescue. She might even create some distractions to help the others get closer to Katara. The Fire Nation soldiers who respect Zuko might be confused by his actions, while those who dislike him might see it as an opportunity to bring him down. And Katara would be using her waterbending skills to defend herself and try to escape at every opportunity.
Zuko's back story made him a very driven character. His banishment and the need to regain his honor pushed him to be tenacious in his search for the Avatar.