I'm gonna write up Korra's. She's the main character of this whole shindig. Unfortunately, I'm not going to be as expansive as I was with the other two simply because I do not have the time to do so, and I've written her in parts with Mako and Bolin's sections.
By going over Mako and Bolin, you already have a rough idea of what her character progression is in the first few episodes. She's the bombastic new Avatar, and the first thing she does as she arrives is break up a crime ring, and then hit one of their warehouses, doing serious damage to their operations, while also forming the beginnings of a strong bond between Mako and Bolin. However, even the beginning, you see Korra's perspectives being challenged. To put it simply, Korra believes she is a superhero, and the bad guys are just bad guys, and all she has to do to fix the world is kick the bad guy's teeth in. Yet already here you have Mako, who is a criminal already being somewhat reformed by her (rather commanding) presence and Bolin, who is a hero that she initially mistook as an idiot. But while these perspectives don't conform to her initial impressions, it is otherwise going pretty much as she expected. She kicked criminal ass, and now was thanked by Lin.
Which is when she finally reaches Tenzin and his airbenders island. It's pretty much unavoidable to explain Korra's character arc without explaining Tenzin's. He's not like Iroh, an unconditionally loving and supportive mentor. Tenzin has his own issues between being on the city council, being the head of the air benders association, and now he's told he also has to train the Avatar. I feel the important part here is that Tenzin's mentorship isn't based in the fact that Korra simply has no other options besides his cranky old ass. If she wanted to learn airbending, Tenzin may be the only airbender left, but there are 3 others who are considered masters of the principles of airbending, even if they can't do it themselves.
No, Korra seeks Tenzin out because she already considers him family. He is the son of Grangran, and her past life. He's been watching over her since she was a child. His kids love her, when they come over. He's just always been so distant. Korra isn't lacking a father figure in her life (family wise, she had a stable and happy upbringing with her average family WHO ARE MOST DEFINITELY NOT SECRET ROYALTY, SCREW YOU LoK), but she expects/needs a certain kind of closeness to the people around her. All her mentors (except for one) became very close to her, and the idea that she goes to train with some random airbending master when she has family for that is just an unpalatable idea to her.
So things go as such: She meets with Tenzin, says she's ready to be trained as an Airbender, but Tenzin refuses. They begin arguing, but Pema comes in and suggests that Korra just came in, and the middle of the night. They're both tired, and in any case they're not going to just send her back to the north pole immediately, so they might as well just welcome Korra for now, eat dinner, and deal with everything else tomorrow. Korra is worried that Tenzin will just force her back. What if he does just refuses to change his mind? How can she be the avatar if she doesn't know the fourth element?
She tries to relax over the next day, playing with the kids, catching up with Tenzin and Pema. Tenzin is mentally preparing for the oncoming headache, knowing that Korra will talk about how great she is and how she deserves to be taught, because she's the goddamn avatar and he has to deal with it. It's not the first time they'll be having this discussion. Then Tenzin reads the news paper and reads about how Korra broke that criminal warehouse. Tenzin kind of panics and demands to know what happened. So Korra essentially explains the events of yesterday (arrogantly proud of her achievement in the warehouse). When she gets to the part about Amon, his powers, and so on, Tenzin's blood runs cold. Korra uses this as her argument. She explains that she needs to learn Airbending to become a fully realized avatar and stop him. Tenzin had heard rumors of Amon, but this is the first confirmation he has of his existence, by the avatar no less. Tenzin considers it strongly. He explains it will be years before Korra truly masters airbending, and Amon needs to be stopped now, which is what he and the police will do. And in any case, if Amon can do everything she describes, they need to get the avatar out of the city as soon as they can. So they were planning on letting her stay a few days, as a small vacation, but it turns out things are serious, and she's leaving tonight. Tenzin leaves no room for argument with the look he gives her.
Korra is devastated. She came here, so many plans. She was supposed to be an airbender, a probender...she was supposed to be the avatar. She stays in her room...turning over the events in her mind. The ship to send her home arrives sooner than she could have believed. At the dock, Tenzin and his family is saying goodbye to her, and then she's on the boat. And the boat is beginning to move...
Which is when she jumps off, saying she's staying. Tenzin repeats himself that he's not going to train her because she wants to be a gungho badass and try to take down a deadly enemy she has no idea how to fight. She tells Tenzin that it isn't about that. If he doesn't want to train her, she understands, but she just can't wait to be the avatar any longer. There is something about Amon that scares her to her bones, but the avatar's entire purpose is to keep the world in balance, and she wouldn't be doing her job if she went off somewhere while the world was in danger. He doesn't have to train her, but she knows this is where she needs to be. She expects Tenzin to continue arguing, but instead he says "About time you figured it out. Your training starts tomorrow."
This is the first real lesson as the avatar that Korra recieves. The fact is that she isn't ready. That's something she's been told all her life, why they stuck her in a gated community so that she can be trained by masters that come to her. It's something that Katara and Tenzin realized was a mistake only when it was too late. For all the fierce personality that Korra possessed, the White Lotus Society's policy had made her passive and sheltered, dependent on the validation of the masters. Korra here was defying the WLS by using her authority as the avatar to declare that there was a problem in the world and she needed to fix it. That was what she was there for, and Tenzin has no right to impede on that. Even though she's under his tutelage, she's ultimately his master, not the other way around. And her defying him like this, for these exact reasons, mean she is ready to take her first step as the avatar and not a super bender that they've been training in a distant land.
So as you might guess by now, Korra has a lot of personality to struggle with. I wanted to keep a lot of the same flaws she had before, but complicate her personality by adding noble traits that she has to struggle with. She wants to be a hero, partly for the sheer glory of it, but also because she legitimately wants to help people. That she is relatively innocent about the true nature of the world doesn't mean she's wrong in that there is a balance that needs to be maintained. There is a lot of growing up she has to do, but her heart is in the right place. As the main character, I don't feel her story arc can be summed up as easily in general terms as Bolin and Mako's who are important but ultimately supporting characters. My vision of Korra would have her explore various aspects of Republic city itself along with development of her relationships with the cast. So it's more complicated. But it's a starting point.
And thus the first tentative Team Avatar is born, composing of Korra, Tenzin, Bolin and Mako, though Korra herself is unaware of it. This would be the last of the 'introduction' episodes, though obviously Asami and Kuvira have yet to make their appearances. However, it sets up the villain, the bigger social issue of the gangs going on, the personalities of the main characters, and the overarching goal. So, not that bad, if I do say so myself.