Education in the Classical Period?

Its hard for me to imagine school functioning differently than it does now, but I know it did so I thought I'd just ask.

I know that education was restricted to those that could afford it, which was generally nobility and monarchy, with most mid-low socioeconomic citizenry having to take up a trade or a family business, but was that really it?

Were there absolutely no schools where the lower class or the poor could pick up reading, writing, and mathematics?

Also, for the upper-middle to upper class, how did their education schedules work? So far as I'm aware people in those days seemed to speak numerous languages, play at least one instrument, be competent in varying degrees in literature and maths, and somehow still had the time to meet social obligations. How in the hell did people (especially kids!) cram in all that learning and sociality while sleeping, eating, and having leisure time?.