The most important chapters to summarize are chapters 1, 3, 8, 18, and 25.
Chapter 1 is important because it introduces the heroine, Katniss Everdeen, and her hunting skills. We also learn her role in the family, and her relationship with Gale. It is an important chapter that introduces the setting and main character relationships.
Chapter 3 is important because it gives background on the political situation in Panem, including the role propaganda plays. The defiant salute and lack of applause at the Reaping is laughed away as a backward custom. District 12’s recruits are major underdogs, and Katniss and Peeta learn what they are in for.
Chapter 8 describes the consequences of Katniss’s trick with the Gamemakers. When they assign her the highest score for shooting an arrow at them, they are making a target out of here. All of the other tributes know her skill, and will want to take her out.
Chapter 18 is important because Rue dies, lighting a fire under Katniss. Katniss’s reaction to Rue is also considered defiance, and one of the things that makes her a threat to the Capital during and after the games. Katniss’s reaction when the rule change is announced that allows for two winners is very telling, because she calls out Peeta’s name. She goes to find him, demonstrating that she does care about him.
Chapter 25 is significant because Peeta and Katniss defy the Capital and the Gamemakers by threatening to commit suicide so there are no winners.
"Hold them out. I want everyone to see," he says. I spread out my fingers, and the dark berries glisten in the sun. I give Peeta's hand one last squeeze as a signal, as a good-bye, and we begin counting. … (ch 25, p. 93)
This seals their fate, especially Katniss’s, and ensures that their struggle is not over. It definitely leaves room for sequels!