What kind of Greek would you like to learn?
Greek is a language among the others. Right?
No, not at all. Greek actually happens to be a more than one languages, including modern, medieval and ancient Greek. And if you ask I would tell that it is not one and the same. Like Latin and Italian is something different.
Well , I do not know if someone want to study ancient Greek. That was the rule in the 19th century when the knowledge of ancient Greek and Latin was a proof of good education. Nowadays education seems to be concentrated on other fields of knowledge, so the interest to learn ancient languages seems to be restricted.
This blog is not in general about Greek. I am not a specialist nor in ancient neither in medieval Greek. I can write about Modern Greek language which is the official language today both in Cyprus and Greece. Diglossia ( η διγλωσσία) was the common situation in Greece until 1976 , when the Greek government adopted Dimotiki ( Δημοτική γλώσσα) as the official language of the state. This means that the legislation is written in Dimotiki and this is also the language in our schools and universities. The previous official language of the Modern Greek State from its beginning until 1976 was Katharevousa (η Καθαρεύουσα: an archaic form of Greek used for official and formal purposes).
I suggest you to read both these articles about Dimotiki and Katharevousa so you can understand better the historical context of the formation of the Modern Greek.
Of course, if you want to travel in Greece and have a conversation with a local ancient Greek won ‘t help nor the Katharevousa (which is practically extinct). I believe that your first choice should be the modern Greek.