There was no mass literacy in the ancient world. Even in Aryan societies. People would have been told poems and stories. Only elites received formal education and a lot of that was language training. Being able to read at all or speak several languages was the mark of an education person pre middle ages. During the middle ages we get scholasticism which was a mix of religious and philosophical dialectic instruction. Scholasticism then becomes several different schools and of course there were schools with secret knowledge (which was code for knowledge that the church would reject). That's where we get the word esoteric.

So your title is a bit off but I see you've corrected yourself in the body. Yes youth would have been told poems and in higher quality families they'd be asked to memorize them. Rote memory was one of the earliest forms or semi formalized education. The rote memory tradition continued with the Christian tradition where non elite 'formal' education meant reading the bible multiple times and memorizing large parts of it.

I'm not sure exactly what poems would have been common. I think you'd need to specify the region and the time period because the poems and stories are going to vary wildy even in Europe. You've also seemingly jumped from asking about what was common in the ancient world to asking what was common in the middle ages. I'd start with German and British stuff because they actually keep semi good records of their own ethnic heritage. Of course we lost a LOT of books after the National Socialists lost the war. Books on all type of European traditions. Western elites used the post war period to privately and publicly purge the west of its past.

https://vho.org/censor/LuedersGB.html

edit. related.

https://www.youtube.com/watch?v=l_fIWlu5zuo