...in Greek, do the rules of grammar apply regarding words or phrases in a series as they do in English? In other words, in English, normal rules of grammar would say that we use a conjunction between the last two words in a series. For example: "Go to the store to get bread, cheese, milk, and eggs."
No, I don't think they do. My sense is that thisἐλθὼν μὲν οὖν πρὸς τὴν ἀγοράν, ἀγόρασον δὴ τὸν ἄρτον, τὸν τυρόν, τὸ γάλα, καὶ τὰ ᾠά.
would not be normal Greek, and that you have something more like thisἐλθὼν μὲν οὖν πρὸς τὴν ἀγοράν, ἀγόρασον δὴ τὸν μὲν ἄρτον, τὸν δὲ τυρόν, τὸ δὲ γάλα, τὰ δὲ ᾠά.
or thisἐλθὼν μὲν οὖν πρὸς τὴν ἀγοράν, ἀγόρασον δὴ καὶ τὸν ἄρτον καὶ τὸν τυρὸν καὶ τὸ γάλα καὶ τὰ ᾠά.
There are, of course, many other ways to say this in Greek.