While I will be arguing the 'pro' side of this debate over the existence of God, I would like to concede from the beginning that God's existence, by its very nature, cannot be absolutely proven except on an individual level. In other words, God can prove His existence to an individual, but because of His nature and will, would not allow His existence to be proven to the world at large, or to anyone who refuses to consider the possibility of His existence.
To understand what I mean, let's look at love. For example, if a man tells a woman that he loves her, it would be impossible to absolutely prove that this is true if she refused to believe it. No matter what he does to prove his love, no matter what evidence of his love he provides, if she refuses to believe he loves her, she will never believe it. Only if she is willing to believe that he loves her, could he prove his love to her. The same is true of God. Only if someone is willing to believe, will God reveal Himself to that person.
However, I do believe that the existence of God is the most logical conclusion given the evidence at hand, that God's existence is far more logical than His nonexistence. If someone is willing to believe, the evidence alone would almost certainly sway that person.
Defining God isn't always easy, and I would even say is beyond human ability, but whatever we consider to be 'God' must have the following characteristics: It must be supernatural, since it has power over nature; It must be an intelligent force, since there is order within the universe and logic to the natural laws; And it must also be a timeless force, since the Law of Relativity proves that time cannot exist without matter, so if God exists outside of nature, God exists outside of time as well.
As for our universe, there are really one two major possible explanations for how it came to exist. 1) It never came to exist, but has existed always in one sense or another. 2) It came to exist at a certain point.
While I (and most of science) believe the second of these explanations to be the truth, I will consider both of them.
If the universe always existed, then it does not require any sort of creator. While this would certainly allow us to reject God as the universe's creator, this possibility would still require there to be some force in the universe greater than nature. The Second Law of Thermodynamics states that, in a closed system, everything in the system will eventually move away from order towards chaos (a 'closed system' is a system in which nothing, neither matter nor energy, can enter or leave). For example, if you were to pour a cup of boiling hot water into a jar, add a cup of freezing cold water, and seal the jar so that nothing, not energy or matter, could enter the jar, the water in the jar would eventually reach a steady temperature somewhere between hot and cold, and would then stay that temperature forever.
While there certainly can be isolated increases in order, for a temporary amount of time, given enough time the entire system would turn into complete entropy. For example, the amount of order on Planet Earth can, and does, increase by natural means, but only due to a decrease in order in another part of the solar system, namely the sun. The energy coming from our sun is moving our sun towards a state of entropy (though very gradually), while increasing the order on our planet. But given enough time, the sun will burn itself out and then the order on Planet Earth will also decrease, causing our solar system to eventually move towards complete entropy.
If, as Carl Sagan would say, the universe is 'all that is, was, or ever will be' (in his book 'Cosmos'), then our universe would be a closed system. Given enough time, then, our universe will move towards complete disorder (what some scientists call 'heat death'). This would certainly take a long time, of course, but if our universe has been around for an INFINITE amount of time already, then the amount of time necessary to bring our universe towards complete disorder would have already passed, and we would be in a state of complete entropy, or heat death, right now.
The only way our universe could have always existed, but not be in a state of entropy, is if our universe is NOT a closed system after all. This would require some force outside of nature to be providing the means of keeping some degree of order in our universe. Therefore, such a force would have to be supernatural, aware of our universe (therefore intelligent), and since time has been proven only to exist within nature, the force would be timeless. A supernatural, intelligent, timeless force. In other words, God.