Elves don't get their first character level until age 120. That means they spend a lot of time until then doing... well... something.
If we look at humans and what they learn in their first 15 or so years, we can come to quite a range of different subjects.
As toddlers, we learn to communicate. This takes quite a few years to do proficiently. Making yourself understandable to your parents is one thing, but talking to other people is quite complicated, and that's on simple topics.
Then at some point we learn actual social interaction with others; not just words, but form and etiquette. How to address people how to bring up subjects, what is acceptable to talk about and what isn't, etc.
When we go to school, we learn the basics of writing, mathematics, history, art, science, our culture, and many other things. Plenty of people learn to write in cursive for who knows what reason, learn to play some kind of instrument, learn painting, maths, algebra, the history of their own nation, other nations, the world, about great people of the past, get some knowledge about basic science and the stuff the world is built out of.
Then after that, we specialize, learning either a profession or maybe to become a researcher which requires indepth study of a very narrow topic. After that, we're ready to explore the carreer world and start doing our professional thing. Or, slightly before we reach that point, we decide "sod it, give me a sword and something to stab."
Now... pull that whole thing into the realm of Elves; creatures of immense grace, almost limitless lifespans, with a culture and history to rival any other. After the first decade or so, your Elf child (they grow a bit slower, after all) reaches their true "toddler stage". They know how to do basic communication, they'd probably be able to pass as a young human, and if they learned their early communication in Common they might be as proficient as a human of that age.
But they're not learning Common, they're learning Elven. And being able to talk to a Human simply will not do. After learning basic communication it's time to be drilled in proper Elven etiquette. They'll need to learn to speak with that unmatched Elven grace and beauty that seems to come so easy to them, because Elves don't usually tell Humans that it takes them 30 years to speak like that. Humans wouldn't understand anyway.
(You know why Elves are always so aloof and arrogant to you? Because you are pissing all over their established etiquette. But they sort of accept that because you don't live long enough to learn it, anyway.)
So now our Elven "child" is already into their early 30s or 40s. They can talk like a true Elf. If you put them in a Human community (and teach them Common, but that's a vulgar tongue, so you don't) they would speak in words like a diplomat but have the wordly understanding of a child. Lots of words, but no substance.
To fix that, it's time for basic education. If you are looking forward to living to 500 and you have a reputation to uphold (remember that the Elven arts are the best there are, always), then this is going to take a while. Not only that, but whenever you learn about a famous Elf (there are plenty) you have hundreds years of history to learn. And your nation probably spans thousands of years, and with the life spans of those involved you probably have really good records of most of it, so there's loads and loads of information to learn.
Then you have art. Learning how to read notes and play the flute is, of course, not worthy of an Elf. Basic education involves learning to play the flute, harp, lute, violin, piano, drums, as well as voice. And proper drilling, too. You need to play each of them excellently. Of course this is all traditional Elven music... it's longwinded and requires patience to listen to; songs taking 30 minutes are no exception. Dreadfully boring to a non-Elven audience and mundane to an Elven audience, so while you can play all these instruments, those crude humans won't appreciate the sheer beauty of your perfected Harmonious Symphony and any Elves you meet can play it just as well as you can.
Then of course, other than instruments you'll also learn to paint, sculpt, write stories and four other disciplines of your choice. Again, all of these are traditional Elven arts (there are a lot of them) which aren't very interesting to other races, but they belong in a proper Elven upbringing. Some of them might even involve works of art that take over a hundred years to fully appreciate. While Humans perform crude arts like making bouquets of flowers, Elves learn to till a garden to perform beautiful flower art on their own, with no interference. As long as you're willing to wait a few years for them to grow, anyway. It doesn't really catch on with most races.
And finally, there's magic and science. After all, an Elf needs to understand both the history of magic, the theory of magic, and the broad applications of magic. Nobody ever tells you that 95% of magic isn't made for combat and adventuring, of course. After learning at the very least a bunch of simple spells and the theory of magic itself, Elves can probably hold an hour-long boring lecture on the theoretical applications of abjuration auras for the purposes of growing stone that lets itself be sculpted more easily than any human expert in that field. If there were any. There aren't, for obvious reasons.
So now that we've had another 50-60 years of basic education, we're approaching a hundred years of age and we have an Elf creature that has more experience than any Human will collect in their life! It's just all fairly pointless outside of Elven society, but then most Elves don't actually leave Elven society so obviously they're being trained to fullfill their role inside of it. The only thing that is left is to train for their assigned profession (perhaps another 50 years of smithing various pieces of jewellery, or maybe another 100 years or so to learn to play a chosen instrument even better, or maybe just more about the lifecycles of various plants so they can learn to plant a field of grain without disturbing a single creature, or another such Elven profession.)
Or they might learn the history and creed of their deity and set their first steps into Divine magic and go out to help the world. Or perhaps they'll learn some utility or combat magic and prepare a quest to recover some forgotten tome of knowledge. Or they'll learn to wield all sorts of weapons and armor and sign up for military duty. Of course they'll approach with that same Elven dedication as any other task and learn all of the involved theory, history, form and everything involved with their future task.
And then, around 120 years of age, they are ready to set out to bring glory to the Elves in the outside world. To an ordinary Human, they seem woefully underprepared for someone their age, but then the Humans don't appreciate the Elves' enyclopedic knowledge (of non-adventuring knowledge), their grace and understanding of language (mostly lost on said Humans, especially when speaking in the crude Common tongue, which takes them days to learn at worst), their theoretic understanding of magic ("just shut down the trap, we don't need another lecture") or their ability to play a host of instruments ("this marching tune takes how long? Don't you have anything shorter?") or all sorts of art ("whaddaya mean it'll take you 6 months to paint me? why on earth would you wait for each individual brush stroke to dry before you make the next?").
Now of course, it would be perfectly feasible to take a young Elf (say, 15 or so), give him a sword, drill him in the ways of combat for 20 years, and then use him as a reasonably trained soldier. Other Elves will consider that child abuse, obviously, so you won't get away with it for very long. Maybe if humans raised it like that, but then you're looking at what (to any other Elf) is basically a pointy-eared Human, culturally. If they ever find a 40 year old Elf who is travelling the country with a band of Humans and getting his adventuring on, he's going to be apprehended, disarmed, and sent back to school where he belongs. And the Humans will (at the least) get a very stern talking to about abusing a child like that.
(Sorry for the length of this, I got carried away!)