Kvothe is curious. Too curious, in fact. He knows things he should never know, as has been mentioned a few times in both novels. Knowing is a power that can shape things, as namers do. Shapers and knowers are at the heart of the story obviously, as Kvothe spends most of his time searching out the numerous questions he has pertaining to his search for the truth of the Amyr and Chandrian.
Every important mystery in the story revolves around knowledge of some sort, whether it be history of a person or group of people, how a thing came to be, or things hidden away for whatever reason.
So let's start with Kvothe. His name means to say, speak, sing, or otherwise proclaim. He wants to know, so he can say, and thus name. He wants the ancient magic which has been curbed and hidden by his professors and by the other forces in the world. He has an unrelenting quest for knowledge. He is also portrayed as resembling one of the Amyr/Ciridae in a few sections.
The Cthaeh is a giver of knowledge (in a tree similar to the bible), albeit spiteful and disasterous to any who hear. The Sithe are Fae caretakers/killers who try and contain this knowledge and destroy any who hear of it. It is mentioned that the Cthaeh and Cinder have bad blood between them, and inferred that he has black eyes because he was bitten or stung by the Ctaeth. Also, when Kvothe shows up at the tree, the Sithe are nowhere to be found. It isn't a coincidence that these things are all linked, and it is possible that the Sithe and Amyr may have similar origin or even be the same. Felurian mentions that none of the Amyr were mortals. The Chandrian are at odds with the Amyr. Some of the Amyr have been potentially converted by Haliax through his power of names, including Cinder, which he illustrates in front of a young Kvothe.
The Chandrian are destroyers of knowledge, but it seems only that which pertains to their own past and signs. They do not destroy children's books, and they do not go after kids or people who sing the fairytale version of their past. Why would they want to hide their reputation? To protect the power of their names that was once commonplace in the past? How many of them were Amyr once? How much disinformation is out there confusing anyone trying to track them?
The Ciridae/Amyr's purpose is "For the greater good" as mentioned by their motto. There are also two sets of Amyr, the original uncorrupted non-mortal TRUE Amyr, and the mortal, corrupt, disbanded version. These two may have coexisted for a time, but when the Chandrian started poaching non-mortal members, and the few remaining non-mortal members felt the walls closing in (if there were any left that is), they went wherever they may have gone (behind the 4 plate door, or lockless door, or in hiding of some sort) and let the mortals run the group for a bit. Obviously mortals can't be more than what they are, and things were shut down eventually, with information about the TRUE Amyr disappearing systematically. Again information and disinformation (as highlighted by Kvothe and Wil's bet and the contradictory stories) is the key to this group.
Abenthy is an arcanist, and the arcanists in general have secrets they are keeping and hiding from the general public. Names and namers are extremely rare, and the information in the library is useless, mostly. Power is wielded by Lorren and the other masters by filtering any dangerous information out, but it wouldn't stop some of the arcanists who had access from leaking information for a purpose. Re-enter Abenthy, who told vital information to Kvothe's parents, setting off the chain of events leading to their death and Kvothe's emergence as the hero.
Now crystallize this all down.
The Amyr want to do what? Destroy/Stop the Chandrian. They seek knowledge and reveal it to their hero to give him the edge against them. If everyone has power, then no one does; the greater good being the flow of information isn't hindered. This has changed over time as the Chandrian have gained an advantage, and now the Amyr are either in hiding, or are extinct/incorporated into the Chandrian.
Chandrian want to stop the spread of knowledge about themselves. They slow the Amyr by capturing and/or killing off Amyr members and try to stop the hero from having the tools to end their grasp on power. They want to keep power by stopping the spread of knowledge and trapping it among a few members.
Kvothe is the spitting image of an Amyr, and intends to ressurect their original purpose as the Hero they were waiting for.
However this ends up in tragedy and disaster, because the secret of power was never meant to be shared...