Your question is a hard one, because I have to assume some understanding of Ancient or Modern Greek to answer (and my Demotiki is rather poor).
Yes, an educated Greek can read the NT in a way very similar to how a modern English-speaker can read Chaucer. I go to a Greek Orthodox Church with several ethnic Greeks in it, and the official Greek text is a Byzantine text-type version still in the Koine. Some of them can read it, but some of them can't. They all get something from it. They can all pronounce everything.
The actual differences in the language can at places be extreme. Ancient Greek has an infinitive and a highly inflected participle. The former was dropped and the latter sharply limited. It had five five cases, but the Demotiki has dropped the dative with the accusative and genitive taking over various functions. The perfect was lost then rebuilt so that in the modern it looks like ours, but in Ancient Greek it was formed by a reduplication of the stem and an ending. It had no helper verb. There are other, important differences, but it's more than I can go into here. You really do need to know at least one for it to make sense.
Learn Modern Greek first IF you have Modern Greeks to teach you. Then you have the advantage of learning it in a way you create an audio path in your brain for it (that's something I didn't get a chance to do when I learned ancient Greek). After that, you can expand into Ancient Greek. This would be the ideal way to do it, because it is a more natural way.
If you don't have a way to really learn it spoken, then there's no use at all trying to master both. Pick up a copy of Athanaze and start studying Attic Greek. You'll get to Koine Greek faster. Personally, I would advise you learn Homeric Greek before any of it if you have the patience. Clyd Pharr's "Homeric Greek: A Book for Beginners" is a wonderful grammar for it. Everyone in the ancient world learned with Homer.
If you decide to go with Attic Greek (don't start with the NT; you need to read more than the NT in Greek to understand it), then get Athanaze. For your biblical studies, you'll need a New Testament AND a Septuagint. The Septuagint was the NT writers' Bible, and studying it will do more for you than any modern scholarly commentary. There are numerous editions of both, and I have no idea what your approach or beliefs are, so I can't tell you which one to get.
After you have a grasp of the grammar you will want to brush up more on your LXX Greek. It'll help your NT significantly. For that there is a good book "Grammar of Septuagint Greek" by Conybeare.
If you can, attend a Greek Church and learn several of the important prayers in Greek. This will also facilitate your learning Though I would advise you choose one with a good hunk of English also, an immigrant congregation with no English at all would simply make no sense to you. You need to be able to draw connections.