A funny thing is that Evangelicals, Protestants, and Catholics who actually know some Greek (as opposed to the “codebreaker types within those religions) answer the question of how to improve one’s Greek with something along the lines of “Read, read, read; and then read some more.”
The Greek Orthodox Church in North America, on the other hand, will teach Neohellenic Greek in their Greek schools (to be fair, I’ve found three Greek parishes in all of North America that teach “New Testament Greek”) to keep the culture alive while the Orthodox Church in Greece commends ignorance of Koine as “Koine has contributed to the “mystery” of the liturgy” The Fathers of the Church would be livid, as anyone familiar with St. Basil can assure you.
Lets us of the Greek Orthodox tradition remember that all Greek—from Homeric to Attic to Hellenistic/Koine to Medæval/Byzantine to Katharevousa to Demotic to Neohellenic—are all very much and absolutely an inextricable part of our Modern Hellenic culture, but more importantly, a part of our Eastern Roman religious heritage.
With that said, here are some suggestions on how to improve one’s Greek that I’ve found helpful and of which I implement:
24 November 2012 – Do You Need to Speak Greek in Order to Read it?
11 December 2013 – Daily Greek Reading Setup
5 August 2015 – Keep Your Greek: Don’t Lose Your Vocabulary
17 September 2015 – 5 Ways to Improve Your Greek Speaking Skills
27 June 2017 – A Strategic Approach to Reading Background Texts of the New Testament (tangent to this is Increase Your Brain Power with Classics)
6 July 2017 – Practice Greek Like a Master Violinist