If you look at the first two chapters the relationship of Katniss to these three characters is clearly explored. In particular, you might want to focus on the first chapter and the response of Katniss to Gale's suggestion that they leave together and survive in the woods:
Leave? How could I leave Prim, who is the only person in the world I'm certain I love? And Gale is devoted to his family.
Gale and Katniss are therefore childhood friends who have learnt to survive in a grim world and are the providers for their families. What is hinted at in this first chapter is that Gale's feelings for Katniss go beyond friendship rather than just being platonic. Katniss is of course responsible for looking after Prim and her mother, as her desire to protect Prim from as much as possible indicates. It is her love for her sister that causes her to substitute herself for the Hunger Games without a thought for her own safety.
With her mother: She isn't quite sure of her relationship with her mother, because of what happened after the mining accident. Her mother keeps "disappearing," and it forces her to find fodd and prepare for herself, as well as her little sister, Prim.
With Prim: Prim is the one she loves the most. She is the only person she is definitely sure about loving, and Katniss would do anything for her. She proves this by volunteering for Prim in the Hunger Games
With Gale: She isn't sure about her relationship with Gale either. She looks at him like an adult figure, even though he is a good friend of hers. Her feelings for Gale become even more confused when Peeta is thrown into the mix.
Katniss's relationship with Gale in her eyes is similar to that between a mature brother/sister pair. They are both just young adults in the story yet they both bear the burdens of their respective families. Because they must both put up strong fronts for their families who rely on them, they can only look towards one another for comfort. This quality is one of the main ways in which Katniss identifies Gale - as the reliable, brotherly figure. In turn, this allows Katniss's personality to still have a soft spot because she knows that even in such a difficult world, there is still someone she can depend on and relate to. In a way, however, the way she defines this relationship with Gale (brother/sister) shapes her attitude towards romance as well, since it is hard for her to grapple with the thought of depending on another human being for reasons other than mutual survival.
Katniss's relationships with Prim and her mother are pretty similar. In both cases, Katniss is the "rock" - the stronger individual both in physical ability (with her hunting) and in terms of emotions. With Prim in particular, Katniss feels a deep sense of responsibility. This explains why she steps up for her sister on Reaping Day. With her mother, Katniss feels disappointment due to her mother's emotional breakdown after her father's death. Yet, as the series continues, her mother earns her grudging respect through her everyday bravery. These two relationships harden Katniss's personality. Because she feels like the protector, she has trained herself to be very stoic and tough emotionally. This explains her fierce determination to not break down at the Reaping and her incredible emphasis on the practical as she discusses future plans with her mother and Prim in the moments before she is carted off to her first Hunger Games.