As most students of Classical Greek know, you can learn the language, read much of the writings available to us, and still have no clue how in the world the ancient Greeks said hello/goodbye. 

However, although not often explicitly taught, we do know some common greetings from the Classical world. Here are some in Greek: 

Ἀσπάζομαι (Aspazomai) - a verb meaning to greet or welcome kindly, can be simply used to mean “Greetings!” or “Welcome!”

More commonly, however, Kaire (Χαῖρε) is used, Kairete when referring to a group of people. This is generic way of conveying greetings, and can be added to with adjectives of amounts, such as Polla (πολλά) which means much, many. 

Other greetings may include sayings such as “hope you are well!” or “how you doin?”

εὔχομαι σε ὑγιαίνειν (eukomai se ugiainein) “I pray you to be well”

Πῶς ἐχεῖς; (Pos Ekeis) “How are you doing?“ 

Πράσσεις; (Prasseis?) “How are you? [Koine]”

And to say goodbye was also often a wish for the good health of the other party, as in:

ἔρρωσο (Erroso) “Be well/strong!” (pl. ἔρρωσθε)

or

εὐτύχει (Eutukei) “Be prosperous!” (pl. εὐτυχεῖτε)

Most information found at this helpful site (also has Latin)