The 7 episodes that mark the evolution Katara and Aang's relationship. : TheLastAirbender
Aang had a crush on Katara since the day he met her. The first time she kissed his cheek was when he returned her mothers necklace that Zuko stole. A family picture of Aang, Katara and their three children. Katara has shown to have a loving and caring relationship with her three children. Katara trained her. The only time that Aang's relationship with katara was threatened was when he trained at the (?) Eastern air temple with the Chi guru.
You must remember, before the fog infects you. I have no family. And Tenzin, oh gods, poor Tenzin. He has the entire burden of the Air Nation on him as the only airbender of the family, and takes his duty so seriously that he stresses himself to the point where he is unable to form a true spiritual connection, clearly a result of his crippling anxiety over the fear of not measuring up.
Remember who you are Tenzin.
You are the son of Avatar Aang. You are the hope for future generations of airbenders. The fate of the world rests on your shoulders. But what if I fail? I am the son of Avatar Aang. I am the hope for future generations of airbenders.
I am the son of Avatar Aang!
Kataang Didn’t Work (and that’s okay)
However, the line that probably sums up the root of these issues comes from Kya: Cutting and running when things get tough! He avoids and evades and takes off to go be by himself so he can gain perspective. We see this multiple times across every medium. And for Aang, there was the weight of a culture on him.
Katara's relationships | Avatar Wiki | FANDOM powered by Wikia
He did have what he felt was a duty to his people to rebuild everything. He also had a duty to Republic City, which as Kya smartly points out, kept him from giving his children the attention they craved. Aang had kids, and yet he was still the Avatar.
Tenzin has kids, and yet still is the only airbending master. A Beautiful Mess This is just so…heavy. Aang runs off, and Katara swallows her feelings. Plus, again, they do like each other. They also both wanted to see the Air Nation come back.
His departure from the Southern Water Tribe to fight in the Hundred Year War had a profound impact on Katara, who felt abandoned and betrayed by him, in spite of the fact that she knew his reasons for departing. Hakoda consoled his daughter by telling her that his departure had a heavy emotional toll on him as well. After becoming wounded during battle, Katara healed her father and decided to assist him in following the troops to the Royal Palace while the other members continued their search for Ozai.
But you've brought it back to life, my little waterbender. Kanna was Katara and Sokka's grandmother. With their mother deceased and their father off fighting in the War, Kanna was the only other relative they had at the time and thus became their caretaker. Katara and Sokka had a habit of calling her "Gran Gran".
It was hinted that despite their strong relationship, being left under Kanna's care did not compensate for her father's departure from the tribe to fight in the war. Katara sympathized with her grandmother's decision, as she also faced the Northern Tribe's prejudices against its female citizens. Katara's mother, Kya, was an extremely important figure in her life, with her death having a traumatic impact on both Katara and Sokka. When they were leaving, Aang remarked that the girls made him feel "like he was at home with his people", making Katara realize that her jealousy had been unnecessary.
This left her feeling guilty for her lack of faith in him, and when he thanked her for agreeing to stay the night, she dismissed his gratitude, claiming that she did not deserve it. She also explained to him that when she saw Kori's family, she saw her future with Aang.
She told Aang to go to a quiet place to think, and he departed with a passionate kiss with Katara. Aang decided that Katara was right. When Aang formed the Air Acolytes from the members of the fan club, Katara watched and admired from afar. Allies On his journey around the world, Aang made many friends from all four nations. Though Bumi remained his only friend from before the Hundred Year War and Aang's one hundred years frozen in the iceberg, he mentioned that he had many friends from the other nations, including the Fire Nation.
Now that those one hundred years had passed, most of his friends from pre-war were deceased, though he had made many new friends after being liberated from the iceberg. Katara, Sokka, Toph, and Suki became his closest friends. He also formed friendships with former enemies, such as Zuko and Iroh. Appa, the flying bisonwas Aang's loyal pet and friend, as well as Team Avatar's primary means of travel.
Aang first met Appa at the Eastern Air Temple when the latter was just a calf during a customary event where all airbender children are paired with a flying bison for life."Halo" - Aang/Katara fanvid (Avatar: The Last Airbender)
After Aang offered an apple to Appa, the young sky bison seemed to bond with him immediately, affectionately licking his face. Appa was faithful and protective toward Aang, and on various occasions he entered into battle to protect his little friend. Aang was similarly protective toward Appa; when Toph blamed the sky bison for leaving a trail of fur for Azula and her team to follow, Aang angrily defended Appa,  and later, when Appa was kidnapped by sandbenders in the Si Wong DesertAang became furious upon finding those that stole him and even entered the Avatar State upon discovering that his friend had been muzzled.
Despite his pacifistic nature, he destroyed all the sand-sailers and almost attacked the sandbenders, had it not been for Katara who managed to comfort and calm him down,  though the grief over losing Appa would stay with Aang for a while, overshadowing his generally cheerful demeanor.
During his time apart from Appa, Aang was often morose and driven to get him back, which he finally did when Team Avatar traveled to Lake Laogai. After reuniting with Appa, Aang cried in happiness. Additionally, he could command him to fly with a cry of "Yip-yip!