Ok, first off Koine, Biblical, New Testament, and Hellenistic Greek are all the same thing. They are the type of Greek in use from about the start of the Hellenistic period through the time of Christ.
Attic Greek is also known as Classical Greek. It was the language in use during the time of Socrates, Euripides, etc.
The oldest form of true Greek is known as Homeric Greek. It is the language of the Iliad and Odyssey.
Modern Greek is in use today and has many differences from all of the above.
Generally speaking the older the Greek the more complicated it is to learn. I'm assuming that the Elementary Greek curriculum you mention is the one put out by Memoria Press. If I'm right then it is Koine/Biblical Greek. If I'm remembering correctly it has 3 levels at 1 level per year. Memoria Press also has a 1-year Biblical Greek program called First Form Greek. After completing it they recommend moving on to Croy's A Primer of Biblical Greek. Croy's book is very grammar based. It is similar to Wheelock's Latin in the way it is set up. The other popular Biblical Greek program is Mounce's Basics of Biblical Greek. It assumes that the student has had no prior exposure to Greek. However, it doesn't stress grammar the same way Croy does.
Good Attic programs for beginners are harder to find, the one I used was called Reading Greek, and I really can't recommend it. It was hard to understand and their were a lot of typos.
The Homeric program I like the best is Clyde Pharr's Homeric Greek course. It is based on the Iliad and has a nice mix of grammar and translation work.
In the end which program you pick really depends on your goal for learning the language.