New idea:
Azula was nine when she decided she was going to kill a Dragon to make her father proud. She cried (secretly) when she found out that her uncle had killed the last two already. It was perhaps unwise, certainly unfortunate, for Ozai to 'comfort' her by telling her she would prove herself in the war.
Azula was ten when she ran away to join the war effort, stowing away on a warship that wound up raiding the south pole. She cried (secretly) when she found out that war was less a romatic adventure that she imagined. It was perhaps unwise, certainly unfortunate, for a fire nation soldier to cut down a water tribe girl in full view of Azula.
Azula was eleven when Katara found her, freezing to death, wandering the ice, lost in more ways than one. She cried when Hakoda told her that a fire nation girl had no place in their village, and helped Azula to take shelter in the nearby wreck of a fire nation warship. It was perhaps unwise, certainly unfortunate, for Sokka to blame Azula for the death of Kya and goaded Katara into running away to join 'her friend the firebender' after Hakoda's departure.
Azula was fifteen when she and Katara found the Avatar...
Could go many ways, but a couple of things seem likely:
Azula may have spent ten years as Ozai's child but she's spent five years with Katara as pretty much her only human contact. So she'll probably be significantly different but still possessive (Katara is hers!) and probably views her early childhood through rose-tinted vision and the war through a bloody-tint.
Sokka hates Azula. Hate hate hate lust hate hate... She is, after all, the only girl he knows close to his age that isn't related to him. Regrets driving Katara away but too proud to apologise even though they have patched things up to some extent.
Katara and Azula are best female friends and fellow benders. Katara didn't sever ties to the whole village but despite Kanna's entreaties, hasn't moved back there, even though she goes there most days. She's a somewhat better bender, having picked up a few tricks from Azula and a little less naively optimistic. Quite possibly has a mild girl!crush upon Azula.
Zuko, as the sole heir, probably doesn't get sent into exile even if he does mouth off. He's got Ozai's attention to ensure he shapes up and probably doesn't find it as nice as he expected. Iroh has been kept at a distance to avoid softening him, with the result that the old man has descended into trying to kill himself with alcohol and other pleasures of the flesh.
Aang, of course, is cheerily oblivious. Without Zuko's presence there is no real prod to send him off, so he leaves the south pole a little later with Azula, Katara and Sokka accompanying him to the Southern Air Temple and eventually the north pole. Sokka would like to find Hakoda, Katara seeks a waterbending master and Azula has a highly unrealistic notion of patching up a peace between Ozai and the Avatar. Once Aang goes Avatar State at the temple, the Fire Nation begin a great search for him, with Zhao as the initial enemy.
From there, unsure.