I'm looking for an ancient Greek word that denotes a trainee craftsman's regime of study under a master, I found the word μαθητεία but I'm unsure if that is a word in ancient Greek or only in modern Greek. Could someone let me know if it is or point me to a better word if it isn't please?
Yes, μαθητεία is a word that would certainly be understood in this context, though in practice it was usual to talk about someone being a μαθητής to someone rather than about μαθητεία in the abstract. This applies not just in the context of manual craftsmanship but for any τέχνη, including e.g. rhetoric.