At the end of Catching Fire, we learn that Haymitch, Plutarch, and many of the previous victors of the Hunger Games are involved in a plot to overthrow the Capitol. We find out that these people know that District 13 still exists, that they have enough control over the Hunger Games to blow up the arena and get Katniss out safely, and, most importantly, that they intend to use Katniss as a symbol (the mockingjay) to garner support for revolution. My theory is that this group's involvement in Katniss's life goes much further back than this plan; They aren't just using Katniss as a symbol of revolution, they purposely crafted her, from a very young age, to be a symbol, with the express goal of starting a revolution.
The conspiracy starts with Katniss's father and Madge's mother. We don't know very much about either of these characters, but what we do know makes them the most likely candidates for the parents of the conspiracy to turn Katniss into a revolutionary symbol. We know Katniss's father is a poacher. This reveals courage and willingness to break Capitol laws, both necessary in a rebel. We know Madge's mother is married to the mayor of District 12, has mysterious headaches during which she sees no one for days, and had a twin sister die in a previous Hunger Games. This reveals a motive for trying to bring down the Capitol (her sister's death), as well as a suspicious closeness to power(her marriage) and behavior that could conceal secret activity (her headaches). These two characters are the same age and both had the makings of revolutionaries, and I think they got together at some point to launch a rebellion. Madge's mother's sister died during the same Hunger Games that Haymitch won, and they both would have watched these Games and seen enthusiasm building in their District as Haymitch got closer to winning the Games. This would then give them the idea to replicate this effect by creating a tribute to win the Games, inspiring all of Panem and giving them the perfect spokesperson for revolution, someone who would be recognized and celebrated by the whole country. They probably realize that District 13 still exists the same way Bonnie and Twill do in book 2, by noticing the same mockingjay flying by every time they show footage of 13 on TV. This lets them know thisfootage is a rerun, and that the Capitol is hiding the fact that District 13 still exists. They adopt the mockingjay as the symbol of revolution because of this, and eventually establish communication with District 13, bringing them in on their plan to use a heroic symbol to garner support. So now they need to ensure that whoever they choose to be the symbol will win the Hunger Games.
There are two ways of doing this: teaching the person to be a good fighter and survivor and altering outside conditions like the arena and the other tributes to give that person an advantage. Katniss's father, being good at survival, takes the first job, and Madge's mother takes the second one, marrying the mayor to become better connected to the Capitol. So they need to groom a kid to win the Hunger Games and become their symbol, and this means they need to train someone from an early age while at the same time ensuring that this young child doesn't tell anyone else he/she is being trained. Because of this, they decide to use Katniss's father's child, who he can train to hunt without fearing she will tell someone. This explains his marriage to Katniss's mother, a friend of Madge's mother and the daughter of a medical family, who can teach this future-tribute the one survival skill he doesn't have, first aid. So they have Katniss, he teaches her how to use a bow and arrow and survive in the wilderness, her mother tries to teach her medical skills (although she probably doesn't know about the plan). These both give Katniss a huge advantage in the arena, and she definitely would have died without at least the shooting and survival skills.
Meanwhile, Madge's mother marries the mayor, gains a greater ability to communicate with the Capitol through him, and brings several Capitol citizens into the plan, including Plutarch and Cinna. She and Plutarch, a gamemaker, design the arena to be perfectly suited to Katniss, woods exactly like the ones in which she has learned to survive with her father at home. It is full of plants whose medicinal uses she knows from her mother (like the leaves she uses for her burns), familiar animals she knows how to shoot and snare, and dangers her father has specifically warned her about (like the nightshade). Through the whole Hunger Games, she almost never encounters anything unfamiliar, giving her a big advantage over the other tributes who come from very different environments. She and Plutarch also manage to rig the reaping so that most of the tributes going into the arena with Katniss are small, weak and young, even one crippled boy (the one from district 10) In fact, the only tributes to present a threat to her are the Careers, who volunteer and cannot be controlled for, and Foxface and the boy from district 3, who are small but turn out to be dangerous because of their intelligence. Finally, they rig the reaping to select Prim. Katniss becomes instantly noticable when she volunteers for her sister, and this gives her an advantage getting sponsors (They never had any doubt that Katniss would volunteer). This is why Prim is picked, even though this is very unlikely statistically, since her name is supposed to only be entered once.
So Katniss volunteers, and before she gets taken to the Capitol, Madge comes to say goodbye to her, not sadly but with a strange sense of urgency. She gives her the mockingjay pin, probably on instructions from her mother, in order to associate Katniss with the symbol of the revolution, the mockingjay, once she's in the arena. She goes to the Capitol and gets Cinna as a designer, someone who is almost certainly in on the plot. This is why he says he specifically requested District 12, and Katniss's revolutionary symbolism would not be as powerful without the eye-catching, fire-themed dresses he designs for her. Appearence is a huge part of Katniss's power throughout the book, and much of it comes from Cinna, like the mockingjay dress he designs in book 2. When Peeta introduces the love angle, Haymitch, who as we know from book 2 is involved in the group, encourages it, knowing it makes Katniss more noticable and sympathetic and gives her broader appeal. He continues to encourage the love act in the arena, and this bears fruit when Katniss threatens to eat the poisonous berries, an action many people interpret as a revolt against the Capitol. (Plutarch and the others would have made sure she was forced to do something revolutionary, even if the thing with the berries hadn't happened. For example, they could have used the moment when she decorated Rue's body with flowers in exactly the same way). So the plot we see in the second book is just the continuation of a much older and deeper plan to overthrow the Capitol using Katniss as a rallying point. This is why they break her out of the arena, and why they spend most of the 3rd book trying to get her to behave like the mockingjay she's supposed to be.