The best way to learn NT Greek is to read the GNT
I'm concerned about learning to read the Greek NT by reading the Greek NT - doesn't that mean that half way through verses I'll remember them and thus just trot our some remembered translation. Wouldn't it be better to read other things in Greek too?
There are significant differences between the Greek of the New Testament and the earlier dialects (e.g. Attic Greek or Classic Greek) and later forms of the language (e.g. Medieval and Modern Greek).
I would be very cautious about learning classical (or modern) and koine Greek together. As a psycholinguist and polyglot I have found significant problems learning/using two (modern) foreign languages at the same time. Learning the two variants of Greek is like trying to learn English and Pidgin at the same time, or German and Dutch - there are lots of similarities and it becomes difficult to keep the languages straight. For example when I am conversing in Dutch, I understand them but they tend to say "sorry I don't speak German". (Note that technically, koine is quite a different language/grammar, notwithstanding the common lexical base, being a creolization based on many dialects of Greek and other languages that developed in the armies of Philip and Alexander.)
But the idea of using koine literature, including the Septuagint (LXX) and possibly other contemporary and later Hellenistic writings is quite appropriate - the best way to learn a language is to read widely in it. But it is best to start with text that is familiar - I've learned many modern languages by reading the Bible in them, and this saves a lot of digging in the dictionary.
The classical language teaching paradigms developed in the late 60s early 70s in Cambridge and Reading are based around only reading actual literature, and avoiding the pernicious construction of artificial sentences by NOT translating into the dead language which nobody really knows well enough to write in or speak accurately. This "best practice" produced a revolution in learning "dead languages".
Ward Powers' "Learn to Read the Greek New Testament" - www.wardpowers.info/LTR.htm - was developed along these lines (Reading school) and when I struggled with Greek I found this much easier than working with the conventional grammar set by the theological college.