"Education is experience, and the essence of experience is self-reliance." (S.5.25)
Right from the get-go, Merlyn wants Wart out there on his own, learning how to navigate the world. You can't do this if you always have someone watching over your shoulder. The training wheels (or here, fins) have to come off sooner or later, because you only learn when you rely on yourself.
Marian showed them how to go sideways, one side after the other; how to stop at once when a bramble caught them, and take it patiently out; how to put their feet down sensitively and roll their weight to that leg as soon as they were certain that no twig was under the foot; how to distinguish at a glance the places which gave most hope of an easy passage; and how a kind of rhythm in their movements would help them in spite of the obstacles. (S.11.32)
Marian teaches Wart and Kay about woodcraft, and how to make their way, sneaky-like, through the forest. This is one of the few examples of a woman participating in the boys' education.
"Well," said the Wart, "I can see why the reptiles who had given up being fishes decided to become birds. It's certainly fun."
"You are beginning to fit things together," remarked Archimedes. (S.18.18-19)
Of course, what's being referenced here is the Theory of Evolution: fishes into birds over the course of millions of years. This reference also signifies that Wart, himself, is evolving in his education.
There was a kind of rushing noise, and a long chord played along with it. All round the churchyard there were hundreds of old friends. They rose over the church wall all together, like the Punch and Judy ghosts of remembered days, and there were badgers and nightingales and vulgar crows and hares and wild geese and falcons and fishes and dogs and dainty unicorns and solitary wasps and corkindrills and hedgehogs and griffins and the thousand other animals he had met. They loomed round the church wall, the lovers and helpers of the Wart, and they all spoke solemnly in turn. Some of them had come with banners in the church, where they were painted in heraldry, some from the waters and the sky and the fields about—but all, down to the smallest shrew mouse, had come to help on account of love. Wart felt his power grow. (S.23.36)
Drawing the sword from the stone is symbolic of Wart become Arthur—moving from boyhood to manhood. At this moment, Wart's hearing the sounds of all the animals that helped him along his way, teaching him valuable lessons. They've all contributed something, and they're present to lend him their strength.
"Tell you!" [Merlyn] exclaimed. "And what is going to happen when there is nobody to tell you? Are you never going to think for yourself?" (Q.2.28)
Merlyn's nothing if not consistent. He wants to get Arthur to the place where he doesn't need to be told, where he can exercise his critical thinking skills and figure things out for himself.
Nobody had told them that it was cruel to hurt [the donkeys], but then, nobody had told the donkeys either. On the rim of the world they knew too much about cruelty to be surprised by it. So the small circus was a unity—the beasts reluctant to move and the children vigorous to move them, the two parties bound together by the link of pain to which they both agreed without question. The pain itself was so much a matter of course that it had vanished out of the picture, as if by a process of cancellation. The animals did not seem to be suffering, and the children did not seem to be enjoying the suffering. The only difference was that the boys were violently animated while the donkeys were as static as possible. (Q.5.59)
This is one of the more disturbing passages in the novels—wanton cruelty to animals. The G-boys and the donkeys both blend together in their descriptions, so you have a hard time telling where the boys end and the donkeys begin. Makes sense, since these boys haven't been educated to know that cruelty is not okay. They don't have a teacher like Merlyn (St. Toirdealbhach isn't exactly of the same caliber), and Morgause couldn't care less about this kind of education.
"[T]he important thing, will to be to catch them young. The old knights, the ones we are fighting against, will be mostly too old to learn. I think we shall be able to get them in, and keep them fighting the right way, but they will be inclined to stick to the old habits, like Sir Bruce. […] That is why I say we must catch them young. We must breed up a new generation of chivalry for the future." (Q.8.7)
Arthur believes the children are the future. He's got to get them young and raise them up in the new ways. Old knights like Sir Bruce? Wasted effort, because you can't teach an old dog new tricks.
His teacher had educated him as the child is educated in the womb, where it lives the history of man from fish to mammal—and, like the child in the womb, he had been protected with love meanwhile. (K.15.8)
Another reference to evolution, this time to the specific principle of "recapitulation," or "ontogeny recapitulates phylogeny" (which has since been disproven). Anyway, the main point here is that Merlyn has educated Arthur through the various stages of evolution, but at the same time Arthur had a loving upbringing (compared to some of the other characters, like the Gawains and Lancelot).
"Don't ever let anybody teach you to think, Lance: it is the curse of the world." (K.27.19).
Because once you start thinking, you become responsible. And that can be a huge amount of pressure for someone to bear.
"Thomas, my idea of those knights was a sort of candle, like these ones here. I have carried it for many years with a hand to shield it from the wind. It has flickered often. I am giving you the candle now—you won't let it out?" (C.14.84).
Arthur is figuratively passing the torch to Thomas. He's finally completed his education, and he's learned that whenever justice depends on exercising force, things will ultimately not be just at all. He sees it as his responsibility to pass this knowledge on to a new generation.