Well come on, she can't stay angry at Zuko forever, because then the situation would be like this:

Katara: Thanks Zuko for teaching Aang firebending and helping him defeat the firelord along with us!... But I still hate you.

Zuko: WAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAHHHHH!!!

Yeah, pretty much. Anyways I think there are three ways he could gain her trust:

1. Not harm Aang or be in the wrong place at the wrong time for 5 episodes.

2. He could save Aang's life or another person's life in the group.

3. (This techniacly falls under the category of theory #2) He could save Katara's life.

What would really gain her trust is saving her life. Because without her in the way then Zuko wouldn't have to worry about "taking a step backwards" in Katara's point of view. So theory #3 is my best bet.

Alright, we know something is going to happen later, she's not going to stay ing the same state of hatred until s/he dies.

Ok, but we know when all is said and done, if Katara and Zuko ever form any hint of a relationship, Aang will be all over Zuko to lay off. If they go even further than that, say a hug or a kiss, Zuko can kiss his helping the Avatar days goodbye.

Anyway, it's obviously going to end up Kataru. Katara loves Aang like a mother, but she likes Haru in a different way. Didn't you see the way she reacted when she saw Haru get off the boat during the invasion? Katara seems to have a liking for older men. Also remember Aunt Wu's prediction - he will be a strong bender. Haru's a realy good earthbender (although still not as good as Toph).

Also, what's really funny and awkward is that Zuko actualy says "Zuko here! ..." When confronting the Gaang. I would've thought he would only have said that while practicing. And besides, don't you notice that Zuko sounds a lot more like Jake Long (Zuko's voice actor's other character) than usual? I don't know, it just didn't seem like the normal Zuko voice - could it be his new good person voice?

It also seems that the writers or the creators or somebody didn't want to have to bring Haru, the Duke, or Teo along, but rather only brought them along just because of logistics - they know it would have been wrong to let the children (if you can call Haru one) become prisoners.

Alright, we know something is going to happen later, she's not going to stay ing the same state of hatred until s/he dies.

Ok, but we know when all is said and done, if Katara and Zuko ever form any hint of a relationship, Aang will be all over Zuko to lay off. If they go even further than that, say a hug or a kiss, Zuko can kiss his helping the Avatar days goodbye.

Anyway, it's obviously going to end up Kataru. Katara loves Aang like a mother, but she likes Haru in a different way. Didn't you see the way she reacted when she saw Haru get off the boat during the invasion? Katara seems to have a liking for older men. Also remember Aunt Wu's prediction - he will be a strong bender. Haru's a realy good earthbender (although still not as good as Toph).

Also, what's really funny and awkward is that Zuko actualy says "Zuko here! ..." When confronting the Gaang. I would've thought he would only have said that while practicing. And besides, don't you notice that Zuko sounds a lot more like Jake Long (Zuko's voice actor's other character) than usual? I don't know, it just didn't seem like the normal Zuko voice - could it be his new good person voice?

It also seems that the writers or the creators or somebody didn't want to have to bring Haru, the Duke, or Teo along, but rather only brought them along just because of logistics - they know it would have been wrong to let the children (if you can call Haru one) become prisoners.