This is where the advantages of Wikipedia are really evident. This will give you access to a broad and varied range of examples of various arguments from Godel and so on, as well as criticism of them. While it is very limited it should give you a springboard to research yourself and explore more compex arguments based on other methodologies such as various forms of logic and Epistemology.
http://en.wikipedia.org/wiki/Existence_of_God
Strangely I suppose, Science. More specifically Lingustics, Hermeneutics, Exegisis and the application of higher criticism, also known as the historical-critical method. You basically study the language(s) how they relate to the historical period, the context, syntax and so on across a range of sources, texts, languages and historical record. Sometimes a single word can alter the entire meaning of a passage, the Pauline Epistles are often subject to such criticism.
Rarely is anything black and white, least of all arguments of Faith or Interpretation. The Bible isn't a single book, it is a collection of texts that seek to teach and inform a particular worldview, in many ways it is exactly a form of lifestyle guide, whether it is outdated or not is subjective and a matter of opinion, but ultimately it is about a message, not about a strict adherence to some literal interpretation of the words, Genesis for example isnt about literally creating the universe et al in a week and making men out of dust, but is an allegory about mans relationship with God and the Universe around him, not a literal 'how to make a Universe...' manual as many young Earthers etc would believe. The earliest Christian Theologians from Philo of Alexandria to Augustine of Hippo expressed this theological position and it is a common and original theme within Christianity (and Judaism, which has no concept of literalism within scripture anyway).
I am not insulted, I am not religious.
Simply put, the relevance of scripture is not what it says about the World, but what it teaches about oneself.