For Ancient Greek, the field of excellent material is at once broader and narrower. There is no stand-out set of materials I would unhesitatingly recommend. Nonetheless, here are a few I have experience with and would recommend.
Randall Buth, runs a number of immersion courses (of which I have no experience), and produces innovative materials for Biblical Hebrew and Koine Greek. If you’re interest is particularly Hellenistic/Koine Greek, this is my number one choice. The materials are well-produced, and begin with picture-lessons with audio, before transitioning in Part 2 to a more dialogue-based and written approach. Buth uses a reconstructed Koine pronunciation, which I understand and appreciate but haven’t transitioned to (for a number of reasons). The materials are a little pricey (especially postage if you’re outside the US), but worth it if you buy the lot up-front. Buth’s materials are available from the Biblical Language Center
For those looking to ground themselves in Classical Greek, or after a more traditional approach, I have three recommendations.
The first is a very hard-to-come-by course by Günther Zuntz, a German Classical scholar with impeccable Greek. The English version of his course is “Greek: a course in classical and post-classical Greek grammar from original texts”, in two volumes (Paperback ISBN 1850757208, Hardback 1850753415). I picked up my copy from my seminary library, and have it almost permanently borrowed. It has excellent original texts, comprehensive grammatical explanations, and exercises designed to instill both verbal and written skills. Zuntz insists you use a tonal accent, and expects you to be understanding metre and memorising verse from chapter 5 onwards.
The second is far more widely available, and is the Athenaze series. It’s in its second edition, and there are 2 volumes. There are also two workbooks available (the first is definitely worth it, the second maybe not so much), and a teacher’s manual (I’m not familiar with the latter). Athenaze would best be described as an inductive-grammar approach. It involves readings, then explanations, and moves from a quaint story of a Greek boy to the broader events of the 5th century history, and selections from ancient authors.
My third recommendation is Donald J. Mastronarde’s “Introduction to Attic Greek”. It is a far more traditional grammar book, notable for its comprehensive approach. Mastronarde figures you should just learn everything, right, from the start. I picked up Mastronarde while doing some 2nd year Classical Greek at university, having stepped into the program without their first-year course (on the basis of prior Koine), and found the JACT course virtually hopeless grammar-wise (the JACT materials make great readers though, see below), and Mastronarde saved me. Subsequently I took a grammar course taught from Mastronarde. Stick with him, and you’ll learn a great deal of grammar. I still refer to Mastronarde for a learner-oriented explanation of certain grammatical features.
Lastly, once you’ve got going, devote yourself to reading. If you’re interested in the New Testament, buy a Reader’s Greek New Testament (the Zondervan one is much cheaper, if you’re short on cash), and devote yourself to reading it. Buy the JACT textbook, and the following readers, and spend time reading a little each day. Whitacre’s Patristic Greek Reader is also excellent. Read, read, read. That is the key to moving from ‘I know some grammar’ to ‘I can read and use my Greek’, avoiding the pitfall of ‘I learnt some Greek but have forgotten everything but the Present Active Indicative paradigm’.