I've posted on English versions of the Scriptures, and on Latin versions. I suppose it was inevitable that I would eventually decide that I had to say something about Greek, since I'm nothing if not anal-retentive.
The text of the Greek New Testament that I think is the best while also being readily available and affordable, is the 27th edition of the Novum Testamentum Graece, edited by Kurt Aland, Johannes Karavidopoulos, Carlo Martini, and Bruce Metzger, published by the Deutsche Bibelgesellschaft (1993; ISBN 3-438-05100-1; for leather binding, ISBN 3-438-05101-7). The text of this edition is identical to that used in the United Bible Societies Greek New Testament. The only difference is that the Aland edition (usually called the Nestle-Aland, since Eberhard and Erwin Nestle produced the first edition in 1898) has a much fuller apparatus criticus, the detailed explanation of alternate readings printed at the foot of each page. Also, if you're interested, there are nice bi-lingual editions that print the Greek text on pages facing the Latin text of the Nova Vulgata (ISBN 3-438-05401-9) or the English text of the Revised Standard Version.
A rather nice supplementary text is the Synopsis of the Four Gospels, available from the United Bible Societies, which prints the texts of parallel passages of the Gospels next to each other on a page, with the Greek on the left-hand page and the Revised Standard Version on the right-hand page (ISBN 3-438-054051). The apparatus for the Greek side prints variant readings for the Greek text; the apparatus for the English side prints variant readings from other translations.
If you don't know Greek but would like to learn it--even if your purpose is only to read the New Testament--then you can't go wrong with Introduction to Greek by Donald J. Mastronarde, published by the University of California Press (1993, ISBN 0-520-07844-6). It does not contain answers to the exercises, so you may want to go through it with someone who already knows Greek, but I don't think that you would have to do that to profit from using the book. There are, of course, "self-teaching" books available, but none of them will be as good as Mastronarde's text in terms of getting you up to speed in the right sort of way to be able to read the New Testament with real fluency, even if they do give you the answers to their exercises.
Once you've gotten into reading the Greek of the New Testament, it won't be long before you start getting into the technical details. At that point you will want to get Manuscripts of the Greek Bible : An Introduction to Palaeography by Bruce Metzger, published by Oxford University Press (1981, ISBN 0195029240). It will show you how to read and date the various papyri, parchment, and other hand-written texts of the New Testament that our present edited texts are derived from. You will also learn a great deal about how manuscripts got copied, transmitted, and, in some cases, tainted.