Ancient Greek

Ancient Greek is nowhere near as commonly taught as Latin, so self-teaching could enter the picture for Ancient Greek. Fortunately this isn't a problem. On one hand (μεν), the most important texts are quite accessible. On the other hand (δε), there are lots of quality materials out there.

Blog posts here about Ancient Greek
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The alphabet
You're going to have to learn the alphabet at some point. The good news is that it is very much like the one you are reading right now, and you probably already know at least a few letters. I know you know π (pi). [Note: I'll link to a YouTube video for learning the alphabet once I make one.]

Beginner texts
I really like the JACT Reading Greek text. It delivers on two fronts. It gives you lots of Greek to read and does it as modified literature readings, which is why you want to learn Ancient Greek. Athenaze (American edition) is much like JACT but with a more cultural bent. I hear phenomenal things about the Italian edition of Athenaze, but I've never seen it to say. Ancient Greek Alive is good on the reading front, but the readings never quite get authentic culturally or literarily. Even so, it's got lots of reading, which is a big plus.

Some people may prefer Koine (aka New Testament Greek) or Epic (aka Homeric Greek), and there are texts geared toward that too. I've not seen any Koine or Epic textbooks that give a lot of reading. And, in complete honesty, if you're going to go through the effort of learning Greek to read the New Testament, you might as well learn Attic Greek. You then can read the New Testament as well as other works—learning Koine will not necessarily equip you to read Plato. If Homer is your goal, go for it. It's easier to work forward from the beginning, if you're so inclined.

Moving on to intermediate
There has been a great number of intermediate readers published via on-demand printing over the last few years. They are an absolute godsend for people wanting to read real Greek. These new commentaries seem to focus on people who want to read the classics in the original language. The vocabulary and generous grammatical help is all on the same page as the text, so there is no need to buy dictionaries or consult outside websites.

Bolchazy-Carducci has also put out some wonderful tools too. They have published two transitional Greek readers. One is for Plato. The other is for Homer. I don't know what they cost, but if you like the LEGAMUS approach you'll like these readers too. They have also published Reading the Gospel of St. Mark, which I have reviewed on the blog.