At my church, I run a class on introductory Greek for bible study. This along with my meagre experience with the language in college (one full semester and some off-line tutoring; you don't need much more than a good attitude to volunteer at church
) hardly qualifies me as an expert, but I can tell you what I've seen.
The
Athenaze Series takes an immersion approach, getting you reading right away and glossing over some of the difficulties. I've spent some time with this book, and personally prefer its approach for beginners, but it may not be your cup of tea. It's also cheap--you can find it for under $5 on Amazon--so how bad can it be? This would be my personal choice.
Hansen & Quinn and Mastronarde (probably the one you saw on Amazon) are other fairly recent texts that I often hear recommended. I've only skimmed these, but it is clear even from a quick look that Mastronarde emphasizes grammar and forms only in the beginning (i.e. very very little reading until deep in to the book). However, if you want a hard-grammar approach, Mastronarde seems to be the way to go.
I personally studied Greek using the old Crosby and Schaefer text. I don't think I'd recommend this, though it is considered a classic. One point I did like: It clearly connected Greek grammar and forms to corresponding Latin ones, which I think is very helpful if you already know Latin.
I will make this last point concerning Greek. Latin is a difficult language, but it is one that can be mastered thru self-study. I personally think this is extremely difficult with Greek, since I found far more things to ask questions about in Greek than in Latin; it is IMO worth trying to find a tutor if you want to improve.