How does the relationship between Katniss and Peeta evolve in The Hunger Games?

Katniss and Peeta barely know each other when the book begins, and by the end they are very close, and possibly in love.

Katniss’s earliest memory of Peeta is when he intentionally burned some bread so that he would have to throw it away and she could have it.  Katniss forgets about this incident until she sees Peeta chosen as the second tribute for the Hunger Games.

To this day, I can never shake the connection between this boy, Peeta Mellark, and the bread that gave me hope, and the dandelion that reminded me that I was not doomed. (p. 32)

Peeta and Katniss naturally bond because they are the two tributes from District 12.  Katniss is suspicious and does not learn to trust easily though.  Peeta takes it one step further by announcing that he is in love with Katniss.  Katniss cannot decide if this is a clever ploy for sponsor money or if he is sincere.

At first, Katniss is angry at Peeta for announcing their relationship.  Then she decides to play along because Haymitch thinks it is brilliant. 

“You are a fool,” Haymitch says in disgust.  “Do you think he hurt you?  That boy just gave you something you could never achieve on your own.” (p. 135)

When the games actually begin, Peeta and Katniss are separated.  Katniss is shocked to learn that Peeta is hanging out with the Careers, a group of well-trained volunteer tributes from wealthier districts.  She assumes he is a traitor and basically forgets about him.

The Gamemakers apparently decide that the Game would be more interesting if it takes advantage of Katniss and Peeta’s connection.  They announce a rule change.

Katniss runs for Peeta immediately.  Now that two tributes from the same district can win, she wants to save him.  This sudden burst seems to show that she really does have feelings  for Peeta.  From them on, she looks after him and even kisses him because she realizes that Haymitch would want her to.  The more affection she shows, the more Haymitch can wine and dine the sponsors, and the more life-saving gifts they can get.  She begins to understand that Peeta has been playing a role too.

…[Hooking] up with the careers must have been a move to protect me.  Peeta, it turns out, has never been a danger to me. (p. 248)

The strategy works.  Katniss nurses Peeta back to health, and genuinely does care for him as she plays for the cameras.  When the rule change is reversed back to the original rule, she convinces Peeta to commit suicide rather than have them kill each other.  They are both declared winners.

Katniss’s defiance was a truly revolutionary act, but at the time she was just trying to stay alive and keep Peeta alive.  However, by doing so she links herself to Peeta irrevocably.  Their fates are intertwined from that point.  She tells Peeta she was just playing a role, but he clearly loved her. 

“It was all for the Games,”  Peeta says.  “How you acted.” (p. 372)

In some ways, she really loves him.  The book ends with her confused more than ever about how she really feels for him.

How do the relationships between Katniss and Gale, and Katniss and Peeta evolve throughout the first book of The Hunger Games?

In the beginning, Katniss sees Gale as simply a friend.  I don't think she wants to see him as anything more.  She doesn't want their relationship to change.  The same can't be said for Gale.  We are given hints that Gale might just harbor something more than friendship for Katniss.  It isn't until Katniss's relationship with Peeta starts to change that she realizes she may have other feelings for Gale as well.  Only then is she able to admit that Gale may have other feelings for her.  Her guilt at being with Peeta and knowing that Gale will see them together shows us how her relationship with him is starting to change.  Just because Gale isn't necessarily around physically doesn't mean her relationship with him cannot change.  Katniss will return a different person and Gale will be different as well.  Of course, it isn't until book 2 that we really see proof of their changed relationship, but we do catch a glimpse of what is to come in book one.

Peeta and Katniss's relationship changes in more noticable ways.  Katniss doesn't really know Peeta in the beginning.  She feels indebted to him from long ago but certainly hasn't harbored the same feeling Peeta has.  During training, Katniss isn't even sure if she trusts Peeta.  She is clearly drawn to him in friendship even though she tries to sort of whether or not he can be trusted.  Once they are thrown together in the arena, everything changes rapidly.  Peeta's love for Katniss develops into something real while Katniss tries to convince herself it is all for the cameras.  Clearly, she has feelings for Peeta as well.  The question is what those feelings really are and how deeply they really go.  Katniss leaves book one confused and uncertain in both of her relationships.

How do the relationships between Katniss and Gale, and Katniss and Peeta evolve throughout the first book of The Hunger Games?

In some ways it doesn't feel like there is much evolution in the relationship between Katniss and Peeta from his perspective.  On the surface, they actually begin to speak to each other and, of course, they develop a bond during the games.  With that being said, their relationship in the beginning was Peeta protecting Katniss despite the consequences (his mom hitting him for burning the bread), and in the end he is still sacrificing himself to protect her.

From Katniss's perspective there is an evolution since in the beginning of the book she was not aware of his feelings and no relationship existed on her end.  By the end of the book, she has developed a bond with him and taken the job of protecting him as well.

How do the relationships between Katniss and Gale, and Katniss and Peeta evolve throughout the first book of The Hunger Games?

It seems to me that the relationship between Katniss and Gale fades away as the book goes on.  We don't see Gale, of course.  Katniss thinks about him some, but she is forced to start to grow more of a relationship with Peeta.  By the end of the book, we are unsure as to what will happen next.  Her relationship with Peeta has grown from zero to something potentially big while her relationship with Gale has become (seemingly) less important.