What, are you nuts? Honestly, that really is the first thing that pops into my head whenever someone approaches me or e-mails me about learning Greek independently. I have no idea whether others can say the same, but anecdotally I put the success rate of such endeavors at less than 5%. By that I mean that in my experience less than 5% of those who launch into learning ancient Greek outside of a classroom setting, even with a very generous interpretation of the word "learning," manage to succed at it. Very few gain the ability to work painfully and slowly through some reasonably lengthy chunk of authentic ancient Greek-- that's essentially how we judge success in intermediate level ancient Greek classes. Once a colleague told me that his success figure would be closer to 10%, but explained that most of the successes were people who wanted to learn Greek to read the New Testament. And that's what keeps me (usually) from blurting out "What, are you nuts?" when people start talking to me about learning ancient Greek without a teacher.
I don't mean that people can only learn to read the New Testament. That text does happen to be written in very plain style to ensure maximum comprehension among its original audience, but it's still an accomplishment for anyone to gain enough Greek to read it easily. My own take is that it's all a matter of motivation. Most people who want to read the New Testament in the original Greek believe that it is the single most important document in the history of the world. Now THAT is motivation.
So my first piece of advice is this: Make sure you want to learn ancient Greek. If you don't, you won't. So go ahead and buy a book if you want to, you might pick up a little bit and satisfy linguistic curiosity--it certainly won't hurt you. But unless you have a reason for studying Greek, that book probably won't get a lot of use. One way of motivating yourself? Read good translations of Greek literature. They will let you know what you're working toward. Trust me, if you love Euripides (or Plato, or Homer, or the Church Fathers, or whomever) in English, you'll go simply crazy for him in Greek where he's so much better. And he really is that good. Most of them are that good.
Next, get a good book: There are a surprising number of textbooks for ancient Greek on the market. Most of them are of reasonable quality. Not all of them will lead to any great success. None of them is right for every learner. I do not have the expertise to recommend books specifically geared toward the New Testament, but of the books for classical Greek, I have three suggestions:
Suggestion #1:
D. Mastronarde's Introduction to Attic Greek
This is a no-holds-barred approach to ancient Greek, written by a professor who got tired of seeing students coming to graduate school with huge gaps in their basic knowledge. You can guess what happened. He wrote a book that filled those gaps. The result is something a bit daunting and very grammar oriented, but of the highest quality. If you are a mature, motivated student, I cannot see going wrong here. On the other hand, I wouldn't recommend it if it didn't have an answer key. Bonus: web-based tutorials and exercises. (This book is also particularly attractive for those with experience in other languages or a background in linguistics.)
Suggestion #2:
The above is a great way to jump right into Greek and focus on reading without having the grammar behind it all remain a mystery. It is one of a set of books, however, and it must be purchased alongside its companion volume Reading Greek : Grammar, Vocabulary and Exercises. Once again this would be hard to recommend without a tool that makes it all even more worthwhile for self-learners, namely An Independent Study Guide to Reading Greek. Another advantage to this volume is that there are several readers in the series designed for those who have finished these beginning books. Generally speaking I would call this series more "user friendly" than Mastronarde's book, but this most definitely isn't "Greek for Dummies" or watered down.
Suggestion #3:
If money really isn't an object, get all of the above. The approaches taken in the two books are very different, but we're talking about the same language here. For self-learners that means you will have two sources to check. Both are indexed nicely, meaning that it's possible to compare, for instance, the JACT volumes' treatment of a topic like infinitives with what Mastronarde has to say.
"But what about this book?" If you already have a book, don't think that my recommendations require you to toss it. As I mentioned before, there are a fair number of books out there, and most of them are of good quality and can be learned from. Another professor of Greek might well make different choices. If you do not already have a book though, PLEASE make sure that you choose one with some means of checking yourself. There is nothing worse than having no idea whether you're making progress. An answer key or independent study guide is absolutely essential.
Fact: Classicists are suckers for those who like antiquity. Use this to your advantage. We certainly don't become professional teachers of Greek and Latin because the pay is great. If you have a college or university nearby, you might try contacting someone in the Classics Department (it may well be hidden as a program in a general Foreign Language Department) who teaches Greek. Although people might be too busy to give you much time, most classicists are unable to resist the lure of an intelligent and interested audience. You will probably be able to get some questions answered and some friendly advice. You may even be able to sit in on a few classes.
Fact: Classicists are strangely attracted to the internet. Who would've thought that a discipline interested in all that old stuff would be so heavily wired (for the internet that is, though caffeine is a reasonably interpretation as well). Of course there are some holdouts who have refused to move beyond IBM Selectrics (you'll get their special Greek type "ball" thing when you pry them out of their cold dead hands), and even some who think anything but good quality paper is an unworthy medium, but as a field we have embraced computers and technology as tools to help us research and teach about antiquity. That means you'll find a fair number of helpful cyber-resources. Use them!
You're not alone. Really you aren't. If you look around enough, you'll find others who share your interest. They may be living next door and part of a reading group, or they may be across the globe but running an email list dedicated to people trying to get some knowledge of Greek. There may not be millions of them, but numbers aren't everything.
Remember, I'm wrong! Sure, my gut reaction to the project of learning Greek on your own is negative. That's partly vanity. As a teacher of Greek I like to think that my kind is indispensable. But we're not. Sure, a lot of people have failed to learn Greek on their own. But people have learned on their own, and as far as I know no one has died because they failed or because they took longer than they had originally planned to succeed. And some have learned quite well. If you have a real yearning, a real desire, it's perfectly possible. Launch yourself into it, and after you have a bit under your belt, buy a volume of your favorite ancient author from the Loeb Classical Library or from Aris & Philips. Both publishers have bilingual Greek/English editions, allowing you to switch between Greek and English in a moment.