Turns out, Zuko's mom never wanted to be with the Firelord, but was forced to because she was a decendant of Roku, and the Fire Sages said a powerful lineage would result from bringing in Roku's family
After being banished, she returned to her hometown, and got together with her original boyfriend.
To leave behind her life, she got a spirit called the Mother of Faces to giver her a new face and repress her memory, and starts living as a woman named Noriko. She later had another kid.
Zuko can't get info out of Ozai, so he gets Azula to talk to him. Azula finds information, but destroys all but one piece, and forces Zuko to let her come with her to search for their mother.
Azula is still a bit psycho on the trip, and sees hallucinations of her mother. She wants to kill her mother because she things she is conspiring to keep her off the throne. After finding her mother, Zuko tells her he will care for her unconditionally, she drops her plan and runs off, leaving a letter she found that she thought would restore her to the throne (apparetly a letter indicating Zuko might not be the Firelord's).
The mother is found, and the Mother of Faces gives her back her face, and Zuko unbanishes her.
And, of couse, 4 seasons of the Legend of Korra.
tl;dw
Aang and Katara have 3 kids: a waterbender, a nonbender, and am airbender. Aang dies young due to the time in the iceberg. Katara teaches waterbending.
Toph has 2 kids by 2 different guys. Teaches metalbending, and eventually becomes the chief of police in a place called Republic City (more on that later).
Sokka goes on to be a leader in Republic City, but has no children we meet.
Zuko stays firelord. Eventually steps down, leaving a daughter in charge.
After the war, Zuko and the Earth King lead the Harmony Restoration Movement that would see the Fire Nation Colonies returned to the Earth Kingdom. But problems develop in the oldest colony, where Earth Kingdom and Fire Nation citizens live together in peace. After a war nearly starts over it, Team Avatar eventually agree to make in an independant city, called Republic City, ruled by none of the nations, where citizens from all over can live. That is the setting of most of the Legends of Korra.