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There are two questions we must address. Is there a God at all? Then if so, is he a personal God that has an interest in us both now and after we die? Or does he just let us live and then go out like a light; allowing us to exist, but only like a tree or a lizard?

Scientists are quite sure that the energy and matter that constitute this universe we live in could not have popped into existence from absolute total nothingness. Therefore there had to be an entity outside this material universe that we know that caused it to come into existence. This entity is what we call God. So He (or It) positively exists, by definition.

The larger questions are: what is the nature of this God, and is anything about it possible for us to discover? Moreover, is it important for us to do so, or is it just meaningless intellectual curiosity?

Could this universe have come into existence all by itself?

Go outside on a starry night. Look up. Every one of those little dots of light is an entire solar system of planets around a sun that is larger than ours. And there are billions and billions of stars and even galaxies that are not visible without powerful telescopes, and who knows how many more beyond that?

Did you create them? Did they just create themselves? Pick up a rock from the ground. This is the kind of thing most of those planets are made out of. Does it look like it could create itself? The ones in my yard don't look nearly intelligent enough!

But, you say, what if there were "first things" that somehow could exist for all eternity, and also had the ability to spontaneously start up the universe, and were somehow great and smart enough to make it wind up so immense and orderly and beautiful, with intelligence and love and everything? Ah, now you have defined God. See? It wasn’t so hard to discover Him after all.

"Can’t the universe itself be infinite, and without beginning or end?”

Your thinking sounds OK on the surface, but that is just not the nature of the universe we live in. Scientists (whose opinion I am sure you respect) assure us that with ever increasing entropy, our universe has a long but limited time to exist. Everything in this universe changes, and everything has a beginning and an end--even stars and galaxies.

The only way out of this is to define God as an entity that exists outside this universe.
Then "God," as we have defined him, cannot have the same nature as the rest of the universe that we see. If he did, our argument is circular and we must still search for a first beginning. Our language has no clear words to describe his nature, we usually fall back on "spiritual." By this we simply mean that he is different from anything that we have direct knowledge of, and that, unlike the world that is temporary and constantly changing, he is capable of being eternal. In fact, logically, he must be eternal. This is the only way he could have brought our universe into existence.

Alternative argument is this:
Assume there is no God: then this world, and happiness in it, is extremely important. Indeed, it is all there is. So it is a great tragedy whenever anyone does not have a long and happy life here. The death of a child, or a painful life in a third-world country, is an unimaginable tragedy, which nothing can mitigate.

If you have seen your own grandchild killed by a car, or have traveled in poor countries and have seen the misery of people starving slowly to death, your own sense of justice demands that there be more to life than the vale of tears we experience in this life. It is just not fair that the only life a small girl knows should end almost before it is begun, in an agony of injuries. It is just not fair that so many millions of people be born into circumstances that they have no control over, and that condemn them to unimaginable suffering every day of their lives.

Are we ready to say that this beautiful universe, which is so incredibly orderly, is also diabolically unfair? That would make us only a great cosmic joke. That is not acceptable.

Our sense of justice demands that these wrongs be righted. But by whom? Not by us, that's impossible: only by a Supreme Being.
We have a conscience that lets us know right from wrong. Yet in this world, the people who do not follow this guide and commit murder and cheat and steal and lie and are selfish and mean are often the ones who are the most successful. Honest, good people living quiet lives of loving and charity often have horrible illnesses, or lose loved ones, or are born into abject poverty. Our sense of justice tells us that there will be, there must be a future time of accounting, when the books will be balanced and the inequities of this world will be resolved. Yet with no God, no Divine Justice, this is impossible.

Many ask: how can there be a God if He allows children to be raped and murdered? We ask: how can there not be a God, in a world where this in fact happens? It is only His Divine justice that can right this wrong. It is only a God with infinite power that could bring any good out of this unspeakable evil.

(Christians know that any child who dies has gotten a free pass. She will be ushered automatically into God's presence and be happy for all eternity.)

And yet... the universe does exist. So your problem is not a true dilemma, only a seeming one.
It is true that scientists are firmly convinced that matter and energy cannot be created or destroyed, only changed from one state into another equivalent state, or transformed back and forth into each other. So a universe that pops up all by itself is scientifically impossible. Yes, scientists are quite right, everything has to be created. But by "everything," their knowledge is of course limited to this universe. We are led to the only possible explanation. Logic and science are both products of this universe. God is not.

So God cannot be a part of this material universe. We define God as the one and only entity that did not ever have to be created, and who brought this universe that we live in into existence from outside of it.
In mathematics we define things in this way all the time. When people only knew of "counting" numbers, there was no solution to a problem like: what is 3 minus 7? 7 minus 3 is easy, but how can you take away 7 objects from a pile of 3 objects? It is impossible. It makes no sense. So we define the question as being the solution. There is a set of questions that are all the same: 3-7, 2-6, 1-5, 0-4 and so on in the other direction. We take the lowest one, 0-4, and define it as our solution. We drop the zero by custom, leaving us with negative four. By now people are well aware of how useful this "impossible" number is. It exists, even if beyond our "normal" thinking.

Similarly, since all numbers multiplied by themselves yield a positive number, it is impossible to comprehend the square root of a negative number. So we make the question itself the solution. We define i as the square root of negative one. Even mathematicians at first thought that this could not really exist, or have any practical value, and so called it "imaginary." No such thing in the universe? Electrical engineers work with i every day, and could not do without it in solving very real problems. All electrical currents depend on it. It exists, even if beyond our "normal" thinking.

Nothing could exist outside of time? We define an entity that simply does exist outside of time, that has to exist outside of time and this material universe, because science knows that matter cannot create itself, or even destroy itself. Something outside of time and the material universe, and not subject to natural laws, has to exist to bring it all into existence. This is what we call God. God exists just as surely as negative and imaginary numbers exist: because they have to. God exists, even if beyond our "normal" thinking.

The difficulties here stem from trying to apply the reasoning and experiences of this world to God, who is not of this world at all. It is a similar situation as the ancient Greeks found themselves in when they tried unsuccessfully to explain Zeno's paradoxes. You will recall the one about how if Achilles gave a turtle a hundred foot head start in a race, he could never overtake it, even if he could run ten times faster. The reasoning went thus: in the time it took Achilles to run the first hundred feet, the turtle could run another ten feet. Then in the time it took Achilles to run that ten feet, the turtle would run another foot. Then in the time it took Achilles to run that foot, the turtle could run another tenth of a foot. And so on and on, to trillionths of a foot. But Achilles would never actually catch up to the turtle, because there is no limit as to how small fractions can become.

Ancient Greek mathematics could not recognize or handle the sum of an infinite series. By reasoning, they had to conclude that fleet Achilles could never overtake a slow turtle in a race, just as atheists must conclude that no one at all made them.