The main conflict clearly centers around the hunger games themselves. The characters experience mostly man vs man conflicts. They must strive to appease the capital citizens and gain their assistance. They must also strive to avoid or kill the other tributes inside the arena.
The characters also experience man vs self conflict as they struggle to decide if and how they will kill the other tributes. They must decide how they feel about killing another human being and then live with those consequences. We see the difficulty Katniss struggles with after killing the boy from district 1.
Finally, the characters experience man vs nature inside the arena. We see Katniss struggle against dehydration as she searches for water. We see Peeta struggle against infection caused by debris and mud in his leg wound. Many of the tributes must deal with injury and illness brought on by their environment.