One popular book that helped me a lot in understanding the history of the Bible and how it was written is Azimov's Guide to the Bible (single-volume version, if you can find it). It's now a little dated, but it was written for a general audience and does a remarkable job in engaging the reader with the biblical text and explaining the views of Bible scholars.
If you don't know biblical languages, I would recommend collecting a variety of translations and compare between them. But I think the most important thing is to read the text fully with its connected, full context. As JWs, we were not really taught the read the Bible this way, but rather jumping from isolated verse to proof-text, which disregards the context. It is much better to read, say, Romans with an eye on how Paul develops his argument and how he supports it and why Paul says the things he does. At the same time, I don't find it as helpful to read translations that do little to break the text into sections and subsections. That is why I like using the Jerusalem Bible version the most, because it takes each text and divides it into sections and subsections and subheadings and makes it very easy at a glance to see where the author is going or when one argument ends and another one begins.
I would also recommend collecting other texts to understand the broader context of the Bible. I would especially recommend the Old Testament Pseudepigrapha and the Apostolic Fathers, but also recommend some of the early church fathers (e.g. Justin Martyr, Irenaeus, etc.). The first will help you understand how early Christianity drew on Second Temple Judaism, the second will show the closest Christian writings to the NT and how Christians developed many issues of faith discussed in the NT, and the third develops the concepts even further. All in all, these sources can show how Watchtower doctrine, which was arbitrarily constructed by texts confined to the current canon, frequently resembles nothing like what early Christians believed.
Finally, getting a commentary (like the Interpreter's Bible) or a collection of commentaries would be another valuable part of trying to understand the text. But you may find that there is no end to this exploration. The text is like an abyss and the deeper you plunge into it, the more you realize that the process of discovery and learning can go on forever (often because of the ambiguities of the text and multiplicity of opinion on how to interpret it -- there often isn't any single "correct" interpretation).
