Many people usually ask me: "is there proof/evidence that God exists?" To someone like me, this is a strange question, akin to asking "is there proof your dad exists?" But again, if they don't know my dad, it's up to me to show them that, indeed, my dad exists.. as strange a question as it may be. But many people are truly stumped by this question, and so here is a quick response I wrote up for someone, with a few edits for polish. I might soon expand on each point elsewhere, but this is the summarized version:
Time is regressively finite, that means you cannot go infinitely into the past because to reach "now" there must have been a finite number of events leading up to it, or else we would never reach "now" (due to the definition of "infinite"). Therefore, there must have been a first event, and that event requires matter. And given the law of non-self creation, or "ex nihilo nihil fit" ("from nothing comes more nothing") there must be an agent outside of the limits of space and time, i.e., infinite and eternal, with the ability to create, that is responsible for that first event. This agent or entity is what we define as "God"... so not only does God exist, but God must exist, due to the existence of everything around us. The question is more a matter of who/what we think this agent is.
Pantheism, which says "god is everything", doesn't work because "everything" cannot be God, since that would limit God to being bound by the limits of creation, i.e., space and time, and by definition God cannot be. Therefore, while creation comes from God, creation cannot "be" God.
Deism actually doesn't work either, but is the safest bail-out for those who wish to reject the existence of the god of any of the monotheistic religions. It doesn't work because it states that God "wound up" creation like a clock, and then let it go off on its own without any further divine intervention. This doesn't work due to many reasons, such as chaos theory and Newton's second law of thermodynamics (entropy), which state that things tend to become more disorganized, and that they are at their most stable when disorganized. As a side-note, even evolution would require a capable, guiding, controlling intelligent entity (or entities) to cause the generation of new information (DNA, for example). But let's forget evolution for now. (I'm a 6-day creationist, though not of the strict 24-hour day variety, since a "day" in the bible is not limited to 24 hours.) Simply put, creation needs God to prevent it from destroying itself. Case in point the fact that we have the nuclear capability of destroying Earth 100 times over. Seriously.
Polytheism doesn't work either unless we have something like multiple persons in a single God, due to the definition and requirement of God being infinite, so it once again falls upon who you understand God to be. The doctrine of the Trinity, for example, (which is probably the hardest doctrine to explain within christianity) is viable, due to its definition as there being only one God, though manifested in 3 persons. (To be clear, the Bible doesn't talk about one God in 3 manifestations, but 3 distinct persons within a single God). But if we were to have multiple gods, then which is the infinite one? Which is the Creator? And if any are "mortal", or can cease to exist, they no longer are infinite or eternal, and hence fall out of the definition of God. So, there can only be one God.
It can be argued that God is personal (as in "having personality") by the fact that creation required design and therefore creativity and intelligence. Intelligence and creativity require some form of personality given the decisions involved in design, and it can't be claimed to have been the result of a random process, because it would not have led to the highly organized nature of all that exists. Especially given biological systems which are irreducibly complex (need all parts exactly as they are in order to function).
So we have an infinite, eternal, personal God with the ability to create, and who is separate from His creation. This leaves us with some form of monotheism.. such as the so-called abrahamic faiths (Islam, Judaism, Christianity).
We, therefore, now face the matter of seeking out what the identity of this creator god is. If indeed God is personal, it would be natural to see or expect some form of interaction with His creation, other than merely maintaining it (since we have disqualified deism above), in other words, the creator most likely needs to be involved somehow to keep it from destroying itself, for example. That is where revelation kicks in. Of course, that's a bit trickier a concept to deal with since this is where the logical, and drier, part of the proof ends and the matter of faith begins. If God has chosen to reveal Himself, then how does/did He?
People who claim to receive messages from God are referred to as "prophets" by those who acknowledge them, and the Bible says that the sign of a prophet is 100% inerrancy when speaking on behalf of God. This means that we can go through the process of elimination to find which prophet or group of prophets seem to exhibit inerrancy by removing those who don't. That means that the Roman Catholic Church, Jehovas Witnesses and Mormonism (Church of Jesus Christ of Latter Day Saints) as well as many other fringe groups are immediately and trivially disqualified as a result of all the unfulfilled prophecies their leaders have uttered on behalf of their God.
I'm not sure what prophecies Islam deals with, but for sure it defines itself as the diametrical opposite of christianity by explicitly stating that "Allah did not beget a son nor was he begotten" so we can eliminate one by proving the other. As a christian, I obviously know more about what the Bible says, so let's focus on that.
In the historical person of Jesus alone, (one quick google shows that) 360+ prophecies found in the Old Testament were fulfilled, of which 190 could only have been fulfilled by Jesus, and no other person in history. Therefore, given the historicity of the person of Jesus, that proves, in one fell swoop, that the God revealed in the Old Testament is the true God, and that we can therefore accept the New Testament as fulfillment of it and therefore truth (unless, of course, we can find that aforementioned errancy in either part... which to this day I have yet to find) and that, therefore, eliminates Islam.
On a personal note, however, I can tell you how dramatically my life has changed since that day the Good Shepheard, who gives His life for His sheep, rescued me from the pit of sin. I remember how I used to graffiti all my notebooks in high school with "life sucks". I remember my suicidal, self-destructive manic depression, and all manner of other problems I exhibited, and I can truly divide my life in half on that day in September, 2001, when He granted me the gift of repentance from my sins and I understood that I was a sinner in need of a savior. On that day I received full forgiveness for all my sins, all my offenses against Him, because they were all laid upon Jesus. He purchased my soul with His very blood, via His atoning death on the cross.
The Bible says that our sin, those bad things we do that offend God, separates us from Him, and while in that state of separation, God does not hear us, and we cannot have fellowship with Him. It is only when those sins are paid in full that we can, and that's what Jesus did on the cross, making it available to all who desire to be reconciled with God, and the Bible says that He will not turn away anyone who seeks Him. For us to receive this forgiveness, we must acknowledge that we are sinners, and repent (which means "to turn away") fully of our sins, understanding that there is nothing we can do to earn this salvation, no good deeds or personal righteousness will ever be enough for us to deserve forgiveness, and that this salvation and forgiveness is the exclusive gift of God, available only through the work of Jesus, the Son of God, and nobody else. It is then that our sins are forgiven in full, and we can enter into that relationship with Him. There is salvation in no other, because if God is infinite, any sin against Him is an infinite offense, and therefore requires an infinite payment. Only God can make an infinite payment, and if Jesus is not God, then His payment is not enough, and we have no hope for salvation whatsoever.