Let me throw my 2 cents in the pond along with the big names.
Besides what I have read for work, study and to follow the news, I do the majority of all my reading each day in Koine Greek; New (1905 book form) and Old Testament and other secular texts, (which I mostly access online).
It took me far more than a couple of months to achieve a truly useful knowledge of Greek! I have done the BA with double majors in Greek and Modern Greek and have read regularly for many years. It probably took me 15 years of working with the language to get a good return on my investment of time and effort in Greek.
By reading widely in Koine Greek, the world around me becomes increasingly more "Greek". English was my first language of education, now after a number of years reading a variety of texts, I am able to understand some of how the world around me works "in Greek". That has made a very big difference in the way I understand a Greek text.
Overall, the biggest difficulty I faced over the years was acquiring a vocabulary in way that got me above beginners level decoding and translation skills. Language learners usually begin by using L2 to L1 dictionaries at beginners’ level then move to L2 to L2(L1) dictionaries at Intermediate level and just L2 to L2 at advanced level. But in this language there are simply no such resources available. There is a difference between initially learning "λύχνος m. lamp", and later reading definition, (simple) examples, collocations and related words like this:
"λύχνος - πήλινον οἰκεῖον σκεῦος καιόμενον ἔλαιον καὶ φαίνον . τὸ φῶς λύχνου ἔλαμψεν ἐν οἰκήματι. ὁ λύχνος φωτίζει σε. ὁ κεραμεὺς τοῦ πηλοῦ ἐποίησεν λύχνον. (That means.. A λύχνος is a household vessel made of clay, which burns olive oil or and gives off light. The light of a lamp lights up a room. The lamp gives you light. The potter made a lamp out of clay.) Other things we can do with a λύχνον are ἐπισκευάζειν / ἅπτειν / (ἐπι)θεῖναι ἐπὶ τὴν λυχνίαν (Trim / light / place upon a lampstand). It is not a λαμπάς (which is for outdoor use). Related words are “wick” ἐλλύχνιον and “lamp-holder” λυχνοῦχος. etc.".
Unless I’ve overlooked an excellent book somewhere, it seems that you’ll have to make your own definitions like that from your reading and analysis – and that is a real drag when you want to transition from beginner (decode and translate) to (upper-)intermediate (lost in the feel of the language and over whelmed by uncertainty). Without such resources, students have to go to an advanced level to provide for themselves intermediate level material so they can progress beyond beginners level (weird!! but we manage).
For grammar, it was difficult for me to get beyond recognising forms and move on to guessing ahead what the Greek would say next, (and realizing what in didn’t say). Teaching grammars don't deal with those things well, and it is something that is left for you, the student to acquire by experience.
That being said, learning Greek is really worthwhile. Depending on what you want from the language, you can get some return on your investment in a few months, but don't just aim for that, be prepared to be in it for the long haul. How long do you spend to read the New Testament in English? A year? Some people say learning a language takes 10,000 hours, if you have a family, then you're young, you have time. Learning the language and reading the book are both real-time activities. Don't be put off by that.
There’s no need to do away with the English translations, they are good.
Why is a child who doesn't "see" arms when he or she draws someone, can't imagine the effect of an action and thinks of most things as concrete entities able to speak languages fluently? What is wrong with my approach to study?