I had some tutoring in it a long time ago using the Wenham textbook - I only remember a few words and the general sound and rhythm of it (pretty) but can sound it out with a little effort because I still remember the alphabet pretty well. (And of course the Cyrillic alphabet I sometimes use is based on it. St Cyril didn't invent it - he came up with Glagolitic, which looks nothing like our alphabets today, and somebody later came up with Cyrillic.)
Let's see if I've got it: first there was the classical Greek of Homer, Plato, et al. (Maybe different versions according to historical period but basically the same language?)
Then there was koine, the language of the New Testament because it was the common second language of the entire eastern part of the Roman Empire, including of course the Holy Land...
...which is why Greek of course was the language of the eastern Roman Empire throughout the medieval era (what historians renamed the Byzantine Empire). The Greek one hears liturgically in the Greek Orthodox Church is this medieval Greek, not classical, nor koine, nor modern.
Then we get to modern Greek, katharevousa and demotic. Katharevousa was an artificial creation coming from the academics' and the government's former pretence that Greeks still speak the ancient language, and was taught in schools until the government gave up on it in 1974. Demotic is what people in Greece really speak, and since 1974 has been the official language, taught in the schools. Same alphabet as the others, though with fewer diacritical marks, but simpler grammar and a very different vocabulary. (The one example I can think of is the word for wine, oinos in the old versions of Greek and katharevousa but a completely different word today.) Right, prodromos, et al.?
I've read that Greeks today can understand classical, koine, medieval and katharevousa about as well or badly as Russians can Slavonic or we can Chaucer.