The answer depends on where you are talking about, when you are talking about, and whom you're talking about. A Greek colonist living in southern Italy in the 6th century BCE, who needed to speak Oscan to the guys ten miles up the road, is a very different case from, say, a Greek merchant living in Hellenistic Egypt in the 300s BCE, who needed to communicate in Latin, Egyptian, Assyrian, and whatever else, depending on his trade partners, or a Roman schoolboy who needed to learn Greek so he could read Homer and Demosthenes.

In areas and time periods without major urbanization, it was often the case that you could walk for a day and reach a village where people spoke an entirely different language from you. Many people would learn multiple languages simply as a matter of course.

There was much more formal foreign language education in large urban centers with either royal courts, temples, or rich merchants who could afford the expense of a teacher and could afford to essentially sit on their ass not actively being productive for however long it took them to learn a new language.

As for the actual teaching techniques, sometimes it's surprisingly familiar. We have tablets from ancient Babylon (2000 BC) with bilingual akkadian-summerian word lists and simple practice dialogs that students had copied out.

We also have some educational materials for Greeks learning Latin- mostly merchants and other people who needed to know if for business rather than for politics or social prestige. Again, there are lots of sample dialogs, especially ones tailored to specific situations one might encounter (e.g., shopping, bargaining, asking for directions). A lot of the texts have the Latin transliterated into the Greek alphabet, because the Greeks learning Latin would have mostly needed to speak it, not write it.

They would also have done rote grammar exercises, writing out or reciting verb conjugations just like a beginning latin class today (amo, amas, amat...). Another exercise was to replace the verb in a simple sentence with every possible conjugation of that verb, for every tense, person, number, and voice.

If you want more details on specifically ancient materials for learning Latin, see "Learning Latin the Ancient Way", by Eleanor Dickey.