Aang and Zuko have a pretty complex relationship throughout Avatar: The Last Airbender. Aang is the prize needed for Zuko to be allowed back into his home at the Fire Nation, and he continuously attempts to reclaim the honor that would come with bringing the avatar to his father.
Aang and Zuko's rivalry was incredibly strong within the first season as Zuko chased Aang across the world, from the South to the North pole. Aang continuously evading the angsty teen, managing to barely escape on several occasions. Throughout their rivalry, a strange bond was formed between the two as they found themselves to be more like allies on multiple occasions. After season one, Aang and Zuko's relationship became even more complex as they struggled to go from enemies to finally becoming friends.
Rivalry Made By Zuko
Zuko and Aang were never truly rivals, at least, not in Aang's mind. Aang just saw Zuko as the boy who never gave up his pursuit of him, he saw him as dangerous. Zuko, however, made up the rivalry and forced it into existence by continuously chasing the avatar and putting all of his own self-worth on capturing the young boy.
So each time Aang and friends evaded Zuko, it was like another punch to Zuko's ego.
Representations of Roku & Sozin
As seen in the flashback of how the Hundred Year War began, Roku and Sozin are meant to be seen as parallels to Aang and Zuko. The two were meant to be friends and Zuko was meant to right the wrong his bloodline had done years before.
Roku and Sozin were best friends, like brothers, but Sozin betrayed Roku and was jealous of the Avatar. Zuko didn't want to make the same mistake. He learned that it was his own bloodline who was in the wrong and desperately wanted to do right by the avatar and become his teacher.
Devoted 3 Years To Finding Aang
Zuko was out at sea, searching for the Avatar for two and a half years before seeing the beam in the South Pole that signified the Avatar's return.
His pursuit of the Avatar across the world added an extra six months to his time, and all in all he had been banished from his home for a total of three years before Fire Lord Ozai accepted him back after his supposed defeat of the air bender in Ba Sing Se.
Never Actual Enemies
Though Zuko countlessly hunted after Aang, he was never actually his enemy. Time and time again, Aang actually saved Zuko and vise versa from other villains. The best example of this is when Aang is captured by Zhao.
Zuko sneaks in as the Blue Spirit and rescues the kidnapped avatar. Of course, he does this with his own best interest in mind, but the way the two work together to escape shows that they have a dynamic untouched by anyone else.
Could Have Been Friends
In that same episode, "The Blue Spirit," Aang had a moment of revelation after he and Zuko successfully escaped the clutches of Zhao and his men. Aang remembered his old friend Kuzan, a boy from the Fire Nation.
Aang described Kuzan as one of his closest friends. He wondered and asked Zuko if, by chance the circumstances were different, the two could be friends. Zuko pauses a moment and then reacts in perfect Zuko fashion, by shooting fire at the young avatar. Of course, later Aangs hope of the two becoming friends comes to fruition.
4 Year Age Difference
There is a four-year age difference between Zuko and Aang throughout the series (not including the 100 years Aang was trapped in ice). Zuko was 16 when the story began, while Aang was 12, and they both were the oldest and youngest members in the series.
Those teenage years are pretty dramatic and each year apart feels like a massive age gap. A 12-year-old is much less mature and way more childlike than a 16-year-old who may think they own the world. This contrast shows between Aang and Zuko time and time again, especially in the first season.
Zuko's Crew Weren't Soldiers
The crew on Zuko and Iroh's ship that he used to track down Aang all those times weren't acuate soldiers of the Fire Nation. They were just regular people who probably, like Zuko, weren't as welcomed in the Fire Nation.
This is why those soldiers probably didn't care as much when Zhao bought them out or whenever Zuko barked orders at them. They thought he was just a snotty prince. This also is probably the reason Zuko's men never did anything harsh against Aang.
Zuko Tried To Bribe Katara
During the episode "The Water Scroll," Zuko tries to bribe Katara into handing Aang over in exchange for her mother's necklace that she deeply cared for. Of course, Katara never would agree to such a bargain, but it is interesting that Zuko thought she would.
This was one of the first signs that Zuko understood the importance of mothers and showed that he understood why this necklace was so important to Katara.
Aang Saved Zuko, While They Were Enemies
Aang saved Zuko's life on multiple occasions, twice in the first season alone, when the two thought of each other as nothing more than enemies. After capturing Aang, Zuko retreated to the snowing cold of the outskirts in the North Pole. He used his fire breath to keep warm and Aang, being an air bender, was able to regulate his body heat. Katara, Sokka, and Yue followed Zuko into the blizzard and, after defeating him, were ready to leave him in the snow.
However, Aang knew if they did so Zuko would surely die from the cold, and so urged his friends to help save him. They brought Zuko back to the North Pole, where he escaped and later attacked Zhao before his death.
Still Wanted To Capture Aang
Even after Aang's multiple attempts at saving Zuko's life, all Zuko cared for was his honor and restoring it in his father's eyes. He continued to pursue Aang, even after being labeled a traitor to the Fire Nation. Zuko was pretty relentless and seemingly never gave up.
After a while, Zuko took a break in his plot to find the avatar and simply traveled with Iroh to Ba Sing Se. After finding out Aang was also in the city, he struggled on deciding what to do with his life, whether he should continue his rivalry with the avatar or if he should open a tea shop and live his days as Lee in Ba Sing Se until his choice was made for him.