"You swim along," said the tench, "as if there was nothing to be afraid of in the world. Don't you see that this place is exactly like the forest which you had to come through to find me?" (S.5.65)
One of the first steps on your road to adulthood is to recognize that the world isn't the safe place you think it is. There are dangers that you've been shielded from (usually by your parents), and here Merlyn (in the shape of a fish) scolds Arthur for not realizing this. Oh, sure: it's all fun and games... until someone gets his head snapped off by a hungry pike! Once you realize there are dangers that you can't see, though, you can start to learn.
The two knights stood to attention while one could count three. Then, with a last unanimous melodious clang, they both fell prostrate on the fatal sward. (S.7.231)
Young people have grand, idealized notions of what it means to be something or do something. How many young boys glamorize being a policeman or fireman? Here, Wart glamorizes being a knight errant, and by showing him the ridiculous fight between King Pellinore and Sir Grummore Grummursum, Merlyn gives Wart a taste of reality. Being a knight isn't all it's cracked up to be. You can get hurt, and sometimes it's just downright silly.
They dragged [Arthur] from under the dead griffin and found Kay's arrow sticking in its eye. (S.12.10)
The Robin Hood episode is not about Wart… but it's about Wart. Wart previously asked Merlyn why he couldn't change Kay into an animal, like he does for Wart. After Wart wears him down on the issue, Merlyn finally agrees to send the pair on an adventure. Here, Kay's the hero, not Wart. But, this episode says a lot about the boy, nonetheless. He understands Kay needs to feel special, and he's learning to be generous towards others and not take all the glory for himself.
The repeating voices in his head, which he could not shut off—the lack of privacy, under which others ate from his stomach while others sang in his brain—the dreary blank which replaced feeling—the dearth of all but two values—the total monotony more than the wickedness: these had begun to kill the joy of life which belonged to his boyhood. (S.13.86)
Does this ant society sound quite modern? Well, it's supposed to. White is modeling this ant society on communist societies of the 20th century. Wart is sickened by how rigid and non-individualistic the ant society is—not to mention their fight over the imaginary boundary that divides two groups that are virtually the same. Growing up, a child needs to learn that s/he has responsibility to the community, but to get too sucked into the community at the risk of leaving behind one's individuality can quickly turn into a nightmare. Once again with Merlyn's lessons, balance appears to be the key.
The Wart did not like to watch Master Twyti for a moment. The strange, leathery man stood up without saying anything and whipped the hounds off the corpse of the boar as he was accustomed to do. He put his horn to his lips and blew the four long notes of the mort without a quaver. But he was blowing the notes for a different reason, and he startled the Wart because he seemed to be crying. (S.16.70)
Adults don't cry, right? Especially adult men. The Wart is surprised that the Master of the Hunt is mourning the loss of his faithful hound. So, the little ditty he's blowing on his horn (traditionally to signal the death of the quarry) does double duty—it's also a song of mourning for Beaumont the hound. Wart doesn't like to see him weeping, since it's outside the realm of his experience. He's learning, though, that part of growing up is messy and involves being vulnerable in front of others.
"I like fighting," said the Wart. "It is knightly."
"Because you're a baby." (S.18.104-105)
Lyo-lyok, the goose, gives Wart a bird's-eye view (literally!) of war. Without boundaries, there is no wholesale fighting. Wart's still a baby, though, and can't see this yet. In fact, he doesn't really understand this fully until the eve before the final battle in Candle.
"Oh, dear, oh, dear, I wish I had never seen that filthy sword at all." (S.23.74)
Sometimes, young people have to be dragged kicking and screaming into adulthood. The sword symbolizes manhood and Wart's need to leave behind the comforts and safety of his family, and start taking care of himself (not to mention the entire kingdom). Getting over fear of change is a big part of growing up.
And he began thinking, stroking his upper lip, where the moustache was going to be. (Q.2.81)
Arthur just got schooled by Merlyn about how the King needs to pay attention to how the poor people are getting slaughtered in his war against the Irish. After all, all chivalry really means is having enough dough to outfit yourself in a metal suit that protects you from getting hurt. The poor people only have their skinsuits—not very protective!
Although Arthur is king, he's still just a young guy—he's still busy coming of age, so he really needs these lessons. Check it: he doesn't even have a mustache yet (and that was pretty important back in the Middle Ages, since facial hair = manliness). Don't worry, though. That mustache will come in time (note the "was going to be"). It's all a process, and the soon-to-arrive mustache is symbolic of that.
Galahad, a priggish, mute little boy, was playing some private game with his dolls—to which he remained attached long after most boys would have taken to soldiers. (K.23.5)
What does dolls versus toy soldier say about how little boys are expected to grow up in this medieval world White creates? How would the dominant culture of 21st-century western society weigh in on this issue?
"You see, I was young, I was nineteen. [...] Everybody told me what a dreadful sin it was, and how nothing but sorrow would come of it, and also a lot of other things about what Mordred would be like if he was born. They frightened me with horrible prophecies, and I did something which has haunted me ever since." (C.4.157).
White doesn't let us in on this horrible event until the last book. On Arthur's coming-of-age journey, trying to drown Mordred (and causing tons of collateral damage in the form of hundreds of drowned babies) is a huge life-altering mistake that he has to work through for the rest of his life. Do you think Arthur tries hard enough to make amends for this? What else might he have done?