Note: "Proving" God exists is really not our purpose. We are only interested in providing clear statements based on facts and logic and then allow the individual decide for themselves who they would like to believe. There have always been people who believed in the existence of God and there have always been those who have denied in His existence. We must realize there are those who will never believe no matter how much proof or evidence we produce. The reason is some people don't want to believe in a Creator or Sustainer. They would not like to consider one day they will have to answer for their actions and for their refusal to acknowledge their Benefactor to whom they owe their very existence. We come to know it is not so much a matter of us trying to convey our beliefs as it is for them to set aside preconceived prejudices against proper belief. Meaning: this is really a matter of guidance from Above. If they refuse even with evident proofs in front of them, this is not between us and them; it is between them and their Creator. Again, it is not our job to "prove" anything to anyone. We only need to present the facts in truth and allow the listener to make up their own mind.
We just begin with simple logic. When something is right in front of our eyes it is difficult to deny it, right? Asking rhetorical questions can be very helpful in presenting our case. Begin by asking the question; "Can you prove you exist?" Yes, of course you can. You merely use your senses to determine what you can see, hear, feel, smell, taste and you have emotions as well. All of this is a part of your existence. But this is not how we perceive God in Islam. We can look to the things He has created and the way He cares for things and sustains us, to know there is no doubt of His existence.
One approach is to suggest simple yet convincing experiments anyone could comprehend. For instance, say to someone, "Consider this the next time you are looking up at the moon or the stars on a clear night; could you drop a drinking glass on the sidewalk and expect it would hit the ground and on impact it would not shatter, but it would divide up into little small drinking glasses, with iced tea in them? Of course not."
Another example is have them consider what might happen if a tornado came through a junkyard and tore through the old cars; would it leave behind a nice new Mercedes with the engine running and no parts left around? Naturally not.
Or ask someone to consider what it would be like if someone told us about a fast food restaurant operating itself without any people there? The food just cooks itself, flys from the kitchen to the table and then when we are done, the dishes jump back the kitchen to wash themselves. This is too crazy for anyone to even think about.
After reflecting on all of the above, how could we look to the universe above us through a telescope or observe the cells in a microscope and then think all of this came about as a result of a "big bang" or some "accident"?