First, it is not modern Greek. If we think about how much English has changed in the last 500 or 1000 years, it’s easy to imagine that Greek has changed at least that much since ancient times.
If we define “ancient” Greek as that used between 700 BC and 100 AD, a span of eight hundred years, there were both regional dialects and changes within that time. For modern students, three dialects are of greatest interest. The earliest is that used by Homer in the Iliad and the Odyssey. There are some grammars and lexica available that focus on Homeric (also called ‘Epic’) Greek. And some proponents argue strenuously that Homeric Greek is the best place to start.
The Greek spoken in Athens in the classical period (5th and 4th centuries BC), often called ‘Attic’ for Attica, the Athenian region, represents the high point of Greek civilization, and is the language of Socrates, Plato, Aristophanes, Lysias, Xenophon, and many others. There are many, many textbooks, grammars, and dictionaries available for Attic Greek, and many scholars recommend learning Attic Greek first, even if one’s ultimate goal is to read the Iliad and the Odyssey.
The third variation is Koine, the common Greek of the Hellenistic world, is the third ancient Greek dialect of great modern interest, because the writers of the New Testament used it. Koine was a standardization and simplification of Attic Greek, and after Alexander the Great, through the Roman period, was the language of commerce, letter-writing, and ordinary life throughout the Eastern Mediterranean. I am not sure, but have the sense that Koine does not vary greatly from Attic.
As should be clear, I am learning Attic Greek, using the highly recommended text by Donald Mastronarde, “Introduction to Attic Greek.” 19 units in, I am very pleased with it so far.