How does Katniss's early encounter with Peeta affect their relationship after they are chosen as tributes?

Katniss's childhood encounter with Peeta makes her more compassionate toward him than she would be otherwise. His kind act as a child also makes her more inclined to trust him, though she does question that impulse several times in The Hunger Games by Suzanne Collins. 
After Katniss's father dies in the mines, her family almost starves. One rainy night, she wanders by the bakery, and the bakery owner's wife screams at her to move on and threatens to call the Peacekeepers. As Katniss leaves, she notices Peeta hiding behind his mother. She's too worried and tired to go far; she sits down by an apple tree near the bakery.
Soon, she hears an uproar in the bakery and the sound of the baker's wife screaming. Katniss expects the woman to emerge to chase her away again. Instead, Peeta emerges and begins feeding a small amount of two loaves he slightly burned to the pig, at his mother's orders. Katniss watches as he checks to be sure his mother is gone, then throws Katniss the two loaves of bread. She says "it was good hearty bread, filled with raisins and nuts."
For his kindness, Peeta received a black eye and swollen cheek from his mother. 
When they're chosen as tributes for the Hunger Games, Katniss thinks that at least with 24 people competing, she won't have to be the one to kill Peeta. Someone else will. She's even worried when he reveals himself to be kind on the train. She thinks that "a kind Peeta Mellark is far more dangerous to me than an unkind one."
Even when they work together during training, Katniss doesn't entirely trust him. She continuously has to stop herself from trusting him because of their past and remind herself that he's playing a game where to win, he has to kill her. When he protects her from the Career pack early during the Games, however, she knows that his persona wasn't false; he really does want her to live. 
Still, Katniss is a pragmatist. She's unwilling to risk herself in a significant way for Peeta before the announcer says that two tributes from the same district can win the game together. Then she yells out his name and searches for him, finding him injured.
This is when their relationship really begins to bloom, and she slowly accepts that Peeta is the kind, generous boy who threw her bread that got her through the hardest time of her life. She battles the other players to get him medicine. She doesn't leave him behind even when doing so puts her in danger. Peeta isn't as skilled as Katniss at survival; however, their past and the popularity their relationship enjoys with the viewers of Panem are enough to keep her by his side.
Katniss tells herself that her feelings aren't real. She believes she's just play-acting for Panem so that they can get more gifts from sponsors enamored with their love story. Ultimately, though, Katniss comes to realize that some of the feelings she's been faking aren't entirely false. She thinks:

And while I was talking, the idea of actually losing Peeta hit me again and I realized how much I don’t want him to die. And it’s not about the sponsors. And it’s not about what will happen back home. And it’s not just that I don’t want to be alone. It’s him. I do not want to lose the boy with the bread.

Her feelings for him—both real and play-acted for Panem—contribute to both of them being allowed to win the Hunger Games, as well as what happens in the other two books in the trilogy. 
If Peeta had not helped Katniss when they were children, it's possible that she would never have trusted him or been willing to work with him as they trained. Without that foundation, their relationship would have never captured the viewers of the Hunger Games, and one of them may have died before the end of the Games.