Have you ever thought about all the things we accept by faith every day? When we cross a bridge or sit down in a chair we don't first take it apart and examine it; we assume it will hold us, just as it has countless others. Nor do we wait until we fully understand electricity before we turn on the lights; we believe in it, even if we don't fully understand it.
In a far greater way, when we come to God we have every reason to believe not only that He exists, but that He cares about us. We see the evidence all around us in the world He created. As the Bible says, "Since the creation of the world God's invisible qualities -- his eternal power and divine nature -- have been clearly seen, being understood from what has been made" (Romans 1:20).
But God has shown Himself to us in an even greater way: by becoming a man and living among us. That man was Jesus Christ. Do you want to know what God is like? Look at Jesus Christ, who was God in human flesh -- and He proved it by dying and then coming back to life. Through Him we not only know God exists, but that He loves us and wants us to know Him.
You can't put God in a test tube and "prove" He exists, the way you can a chemical; God is far greater than that. But you can know Him, by giving your life to Jesus Christ -- and that is the greatest discovery you'll ever make.