If you love philosophy of religion like me, and like to wander the stacks in the 100/200's area, then you love reading about arguments for the existence of God, the rebuttals, the replies to the rebuttals, etc. It all begins with Saint Thomas Aquinas. In only a few pages, he gives us his famous five:
- The First Mover: everything is moved by something else. The tree was moved by the wind which was moved by the weather which was moved by something else, and so on. This could either go on to infinity, or it could stop with a "Prime Mover," a being that gets the ball rolling. That's God. Aristotle, a Greek philosopher that was not a Christian, believed in a Prime Mover (Thomas actually snatched the argument from him).
- The First Cause: everything that happens is caused by something else that usually comes before it. What caused you?--your parents, their parents, their parents, and so on. Because every physical event must have a cause, this could either go on to infinity, or it could stop with an "Uncaused Cause," the beginner of the Big Bang so to speak. That's God. Check out Dean Overman's book for a current example.
- Contingency: When I was a kid I remember sitting on the couch thinking: what if nothing existed at all? No universe. What would that be like? I closed my eyes and could only picture black space, but then I thought to myself: black space is not nothing, it's something! I couldn't imagine or even think about it; it was such a shocking thought. When we look around we see things that pop into existence and then die. They never had to be in the first place. What if everything was like that? If nothing has to exist, then we can imagine at time when nothing exists--no matter, no space, nothing! This is impossible because you can't get something out of nothing. Therefore there must be at least one thing that must necessarily exist. That's God. Check out Paul Davies "fine-tuning" argument in Cosmic Jackpot: Why Our Universe is Just Right for Life.
- Degree: we use terms like "good" and "honest" and "noble" that point to some standard of perfection, some benchmark. When we say a person is honest, we are saying they have some degree of that virtue. There must be a concept of perfection, which helps us to know this. That's God.
- Teleology (Design): everything seems to be directed towards some goal, or end, or purpose. Even ants build complex houses, and everything seems to work together. The orchestrator behind all the design is God. Francis Collins, the DNA guy, has a similar argument in The Language of God.
Although these are Christian arguments, they are used for other monothestic religion (Islam, Judiasm) and probably others (they began as Greek arguments). The history of these five arguments is incredible; they have been transformed, altered, defended, rebutted, discarded, revived. Philosophy of Religion and Karen Armstrong's The Case for God will give you a good overview. Also don't forget Blaise Pascal's argument that, if you were a betting man, you should at least bet on God. And you must read William James's Varieties of Religious Experience, a very nuianced and pragmatic argument.
As for rebuttals, a good start would be The Atheist Debater's Handbook, 50 Reasons People Give for Believing in God (author doesn't think they're good reasons), God: The Failed Hypothesis, and The Portable Atheist.
Aquinas Shorter Summa