Rebbi Yehudah held bad things come into the world on account of amei haaretz. What is the minimum amount of Torah one would have to know to not be considered an am haaretz? Who would this apply to if they didn't know this? For example children would likely not be called this since they are still learning? My assumption then is the term applies after a certain age; is this correct? What if even after a certain age that person is actively engaging in learning Torah to make up for not knowing before? Would they still have this title and if so what is the minimum amount they must know to lose this title?
The Gemara in Sotah 22a says one who learnt Chumash without Mishna is a Boor (lower than an am haaretz). One who didn't even learn Chumash is worse. The Mishnah Brurah 199 Bais says a "complete" am haaretz does not even know Chumash and Mishnah.
The Gemara in Brachos 47b (as well as in Sotah 22a) brings Achairim who says one who learnt Chumash and Mishna but did not [Rashi] learn Gemara is an am haaretz. The Beniyahu (Ben Ish Chai) there asks: someone who learnt Chumash and Mishnayos but did not get up to learning Gemara yet is an am haaretz? He answers that the Gemara meant that he does not want to learn Gemara, he says Mishna is enough for him. But someone who just didn't get up to learning Gemara is Kosher and proper.
The Beniyahu then explains the next gemara according to this answer but as for what concerns this, he says, the Amorah thought that if at the other Amoras age he did not learn Gemara yet, it must be the reason was because he did not want to learn.
I feel that this seems to answer most of your questions. Children are not am haaratzim because they are not up to Gemara. One should at least know Chumash and Mishnayos (I don't know if it means all, most or some) It would seem from the last story that based on your age you need to know the amount of Gemara expected of someone at that stage/age.
A slightly different answer based on my learning and understanding. Don't let the fact that I am a Rabbi lend any authority to this answer, it has just as much chance of being wrong as any other answer based on reasoning and experience.
The Gemara in Brachos 47b (as well as in Sotah 22a) brings Acherim who says one who learnt Chumash and Mishna but did not serve Torah scholars is an am haaretz.
Another Gemara in Pesachim 49b describes an am haaretz as someone who is only out to satisfy his own need, in a context where serving another is the actual mitzva.
So combining these two ideas, as well as ideas from mussar and chassidus, I would posit that an am ha'aretz is not necessarily someone who hasn't learned a certain quantity of Torah, but someone who has failed to absorb a particular lesson in Torah.
What is that lesson?
The lesson is that serving others is good. The yeitzer hara and even the nefesh bahamis are naturally geared towards self-service as the ideal, whether it be because of the seduction of pleasure therein, or the simple logic of "I must look out for myself and survive".
Torah teaches us that there is another side of life, and that is serving others. It is a greater good. There's a pleasure in it that is far greater, as well as the fact that it is a Godly, super rational endeavour that emulates Hashem. Much more to be said.
An am ha'aretz is someone who hasn't figured out that latter lesson, despite a lot of Chumash and Mishna knowledge. A little bit of serving Torah scholars [and perhaps one's spouse], or Gemara learning [Rashi] in addition to this seems to be the required method to lead out of being an am ha'aretz, as insodoing one will come to actually experience tangibly the concept of serving others l'tov, and the prior learning of the Chumash and Mishna about the importance of this will actually penetrate the heart.
How much learning is required to bring all this together for a person will vary from person to person, and I defer to Lanied Sofey's answer for that.