Why think God exists? Here are a few of the best reasons, in my studied opinion:
First and foremost for me are the ancient Jewish and Christian writings. I am referring to most, but not all, of the writings collected in the Protestant Bible. There is an important principle that rational people live by. We may call it “the principle of testimony.” In short, rational people accept the testimony of others that X or Y happened unless they have a good reason to doubt the testimony. Put another way, rational people rightly place the burden of proof on showing testimony is false, not the other way around. In other words, rational people do not go around doubting everything people testify to. Just the opposite. Rational people accept what people testify to unless there is a good reason not to (e.g., the person testifying has a history of lying, or the person testifying could not have been privy to the things they are testifying to). Given, then, the principle of testimony, it seems to me that we should accept at face value the essentials of the history described in the ancient Jewish and Christian writings. But what does this history describe? It describes actions-words and deeds-of God, that is, of Yahweh. So given the principle of testimony, it seems to me we should accept the testimony of these writings that a being, Yahweh, did and said the things the writings say he did. This means that, given the principle of testimony, the ancient Jewish and Christian writings constitute one reason to believe God exists. Moreover, the advantage of this evidence from the ancient Jewish and Christian writings is that they not only constitute evidence that some God or other exists. Rather, they constitute evidence that a specific God exists, namely, Yahweh. So these writings not only give us reason to believe God exists, but they give us reason to think that God is like the God, Yahweh, described in the writings.
Now it is not only because of the principle of testimony that I accept the testimony of the ancient Jewish and Christian writings as, in the essentials anyway, true testimony. There are two main other reasons I accept the testimony of these writings as true. First, there is the fact that the writings are (very) unflattering to those who wrote and passed them down, that is, the Jews and Christians. The writings of the Hebrew Bible (Old Testament) are extremely embarrassing to the Jews inasmuch as they constitute essentially a record of how disobedient and stiff-necked the Jewish (Hebrew) people were almost all the way through the history. Similarly, the Gospels of the New Testament are very unflattering to the first Christians. They even relate how Peter denied Jesus, not once, not twice, but three times(!) during the most crucial hour when Jesus was betrayed. People making up things do not tend to make up things embarrassing to themselves and/or those closest to them. So it stands to reason the ancient Jews and Christians who wrote and transmitted these histories did not make them up. Second, the God of these ancient testimonies agrees with the God of the most elevated reason, and this is in contrast to the surrounding conceptions of God at the time these documents were penned. Indeed, the God of these ancient records stands in stark contrast to the conceptions of God of any time or people. But how explain this if it was all made up by the authors of these documents? If that were the case, then we would expect an un-elevated, even crude, conception of God in keeping with the conceptions that human beings have always come up with apart from these writings. So these are two further reasons, in addition to the principle of testimony, that I accept as essentially true the testimony of the ancient Jewish and Christian writings to the fact that there is a God and this God is Yahweh.
In addition to the testimony of the ancient Jewish and Christian writings, I believe there are yet other good reasons to believe God exists.
For example, there is the fact that if the universe is a merely possible thing, i.e. something that could exist and also could not exist, which it very much appears it is (see below on the beginning of the universe), then it is more rational to believe that there is a reason the universe exists rather than does not exist than it is to believe there is no reason for this. And ultimately that reason must be that a being who is not a merely possible being, but a necessary being (i.e. a being who cannot not exist), must be causing the universe (a merely possible thing) to exist. Otherwise, one must posit an infinite regress of merely possible beings causing the existence of the universe, but then there is ultimately no explanation for why any of the merely possible beings exist. Put differently, it seems irrational to think something merely possible (the universe) exists without being caused by a necessary being. So it must be that the universe exists because it is being caused to exist by a necessary being.
Similarly, if the universe had a beginning, which all indications from science are that it did (hence the currently prevailing Big Bang cosmology), then it seems more rational to believe something caused the universe to begin than to believe it just “popped” into existence somehow. No one has ever witnessed anything just “pop” into existence, so there is no good reason to believe such a thing could happen. Moreover, it just seems self-evident to me, and to most others I would think, that anything that begins to exist has a cause of its existence. But what would be this cause? Clearly something that does not begin to exist but is eternal.
Now the two lines of reasoning for the existence of God just given do not tell us as much about what God is like as does the evidence from the ancient Jewish and Christian writings, but still they tell us quite a bit. They tell us that the being (God) that caused, and continues to cause, the universe to exist is an eternal, necessary being, i.e. a being who cannot not exist, in contrast to the merely possible beings (beings that can and cannot exist) that this being causes to exist. Moreover, they tell us that this being is immensely powerful, seeing that he can cause things to exist ex nihilo (out of nothing). Further, they tell us that this being is incredibly intelligent. This we see from the complexity of the things he has created. Finally, they tell us this being is surely good, given all the good things of the universe and how the things in the universe are adapted to each other to bring mutual pleasure to each other.
This leads me to another evidence for the existence of God, namely, the apparent design of the universe. In short, it very much appears that the universe has been designed. But design implies a designer. Hence, the apparent design of the universe is one more piece of evidence for the existence of God. This is actually the most ancient and historically widely-esteemed argument for the existence of God. The Apostle Paul (Saul of Tarsus) even presents this argument in his most important extant writing, namely, the Epistle to the Romans. This argument has lost some of its luster in recent centuries do to evolutionary theory, but this is without good reason. For the theory of evolution (macro evolution) is questionable at best, and, further, even if it is true then macro evolution itself appears the product of design and so demands a designer to explain it. Also, there are other elements of the universe that the theory of macro evolution cannot account for and which appear to be designed. The fine-tuning of the universe for life stands out above all. In short, according to scientists who study these things, the universe is fine-tuned to support life. If certain facets of the universe were even slightly different than they are, then, it is my understanding (i.e. the scientists say), the universe would not be life-supporting. So this appears to be designed.
Finally, to give just one more evidence for the existence of God, consider the evidence that Jesus of Nazareth was raised from the dead. In my studied opinion, there is no good alternative explanation to account for the evidence. The principle of testimony, as already explained, tells us that we should accept the claims made by Jesus’ earliest followers that Jesus was raised from the dead unless we have good reason not to, so this is one piece of evidence. But even without accepting the claims at face value, if one accepts only the minimal facts that the majority of experts (Christian or otherwise) of Christian origins accept surrounding the possible resurrection of Jesus (e.g. the empty tomb), they seem to me to lead inexorably to the conclusion that Jesus really was raised from the dead as his earliest followers claimed. In other words, the best explanation of a number of (minimal) facts that virtually all scholars who are experts in the area accept seems to be that Jesus of Nazareth really was raised from the dead. But if Jesus was raised from the dead, then this is evidence of the existence of the God who raised him from the dead. (Indeed it is evidence of more than this. It is evidence that Jesus was/is someone very significant himself.)
Those, then, are in my opinion the main reasons to think that God exists. They are the chief reasons I believe God exists, and I believe they are more than ample reason that any rational person should believe God exists.