How much if anything did tywin teach cersei? I always got the impression he didn't bother (focusing on jaime instead), especially as every time I can remember that cersei thinks "I'll do this, it's what tywin would have done" it's directly contradicted by things tywin has said, in her presence no less! What things does it seem she has definitely picked up from him?
I think there are two parts to this. First, without PoV access to the inside of his skull, Tywin’s not necessarily a reasonable source of information on Tywin’s motivations or explanation for why he did things one way and not another. He’s more a rationaliser than he is actually rational, and a great deal of his actions during ASoIaF are brutal overreactions based in his own sense of superiority and fear of being laughed at. (The image of super cool, super smooth, hypercompetent diplomat/politician/general Tywin can go die in a privy and rot on its bier. Oh wait that was GRRM’s point.)
Second, Tywin doesn’t have to be actively trying to teach Cersei anything for her to learn things from him. Cersei, Jaime, and Tyrion all looked to Tywin for example and model themselves after him. What Cersei in particular took away from Tywin is a warped idea of what strength is, both emotional and political. And the warping is inherent to Tywin’s example, not peculiar to Cersei.
The command came easy to her. My father had steel in his voice as well.
Within the tower, the smoke from the torches irritated her eyes, but Cersei did not weep, no more than her father would have.
- Cersei I, AFFC
Jaime also says that Tywin taught him that “tears were a mark of weakness in a man.” (Jaime I, AFFC)
The point is, a steely voice and dry eyes are way down on the list of things that denote emotional strength - in that very Cersei I scene, for instance, Kevan attempts to pray and fails, overcome with grief, and yet he’s back on his feet shortly afterwards. The only person who thinks less of him for needing a moment (in this not immediately dangerous situation) is Cersei. Then there’s this heart-chilling moment:
[Tywin] never wanted love, though. “You cannot eat love, nor buy a horse with it, nor warm your halls on a cold night,” [Cersei] heard him tell Jaime once, when he had been no older than Tommen.
- Cersei II, AFFC
Did Cersei not learn from this teachable moment? I’d argue that Cersei probably learned more and better from that than Jaime did. There’s another flashback to Cersei learning from Tywin in that chapter as well, as she watched her father at a feast.
Aerys and his lickspittles laughed loudly, whilst Father stared at Rykker over his winecup. Long after the merriment had died that gaze had lingered. Rykker turned away, turned back, met Father’s eyes, then ignored them, drank a tankard of ale, and stalked off red-faced, defeated by a pair of unflinching eyes.
[…] It is my look they must flinch from now, my frown that they must fear. I am a lion too.
- Cersei II, AFFC
See also Cersei X. As she’s arrested, Cersei shouts “I am a Lannister, unhand me!” Not to mention
She was not meek Margaery Tyrell, to don her little shift and submit to such captivity. I will teach them what it means to put a lion in a cage.
- Cersei X
This ends up with Cersei in an even worse shift and without either anything to drink from or relieve herself in. The emphasis on Lannister pride is certainly a Tywin-taught belief. It’s part of the first impression we get of him:
“By my lights, it was you who started this,” Lord Tywin replied. “Your brother Jaime would never have meekly surrendered to capture at the hands of a woman.”
His father ignored the sally. “The honour of our House was at stake. I had no choice but to ride. No man sheds Lannister blood with impunity…”
- Tyrion VII, AGoT
[Qyburn] gave her an apologetic smile and told her of a puppet show that had recently become popular amongst the city’s smallfolk; a puppet show wherein the kingdom of the beasts was ruled by a pride of haughty lions. “The puppet lions grow greedy and arrogant as this treasonous tale proceeds, until they begin to devour their own subjects. When the noble stag makes objection, the lions devour him as well, and roar that it is their right as the mightiest of beasts.”
“And that is the end of it?” Cersei asked, amused. Looked at in the right light, it could be seen as a salutary lesson.
- Cersei V, AFFC
No traceable line to Cersei’s beliefs here at all.
I’m also sure Cersei got her first lessons in misogyny from Tywin (see above, too). Tywin’s most outrageous atrocities throughout ASoIaF are directed towards punishing women who had got the better of him/the Lannisters somehow. Tywin here models extreme, and often sexual, violence as a good way to deal with ambitious, scheming women. The very fact that Tywin didn’t bother to formally educate Cersei in politics taught Cersei something about misogyny too.
Politically, it’s a huge weakness that Cersei buys into the myth of Tywin the hypercompetent. Her obsession with being her father’s true heir is as damaging to her as it is to Jaime and Tyrion, and bad for the kingdom too.
No one had ever balked her lord father. When Lord Tywin spoke, men obeyed. When Cersei spoke, they felt free to counsel her, contradict her, even refuse her.
- Cersei V, AFFC
This is what she’s trying to live up to - total unquestioned obedience, because the lesson she took from her father is that that was what was necessary for political strength. Even if it’s not what he meant to teach her, he did.