I suppose you could call me a semi-gnostic Catholic here. I believe there are things we can and can't understand about God due to the entire nature of God being beyond our understanding. I also believe that proof of God's existence is entirely in the hands of one's individual experience. For each person, their evidence, or lack thereof, of God's existence will be different, and it will probably be more circumstantial or ancedotal evidence, experiences one can recount or write down, rather than a thing that can be scientifically analyzed. My own proof of God is something I will never be able to explain unless someone has lived my life and had my exact experiences. Likewise, I believe this to be true for anyone. One finds their own belief system through their own life's journey. I don't believe in something just because it feels good or it sounds cool. I believe in it because it has a major impact on my own life in some very meaningful way. And that's not something that someone can simply wrap their head around and rationalize. Belief, faith, transcends human logic and also human emotion. In a way, one could say it is a combination of the two into some conscious idea that you have such strong certainty in that you are willing to defend it. And I believe that if you truly believe in something, you should also believe that it is an idea worth defending. If you can't muster up any sort of defense of your idea whatsoever, did you even believe it in the first place? I would hazard a guess at no, no you did not.
But one thing I do like about the Catholic Church is something that Benjamin Moore pointed out above, that it contains over 2000 years worth of tradition and faith, and yet it is remarkably open to change and dissent when compared to some of the Protestant faiths today. The Church, especially since the Reformation and the Renaissance, is ever open to new knowledge and discovery, and as new things about our world are brought to light, our understanding of the world, our place in it, and God's plan for it is also ever changing. We don't believe that the Bible should be taken absolutely word-for-word in a literal sense, but in a sense of moral guidance and the real deeper meaning behind what the words say. The Bible is not a science book, nor is it a history book. There are some historical facts that can be found from the Bible, but one should not read it to learn a history lesson. God did not create the world in seven sets of 24 hours. It took thousands and even millions of years to do so, and the Church understands this. We are even relatively accepting of the idea of evolution. I, for one, believe that God created the world and evolution was how he did it. But anyway, that's how I go about things. Learn and understand what is possible to be grasped by the human mind. The rest is up to your own experience. My experiences tell me that God exists. Yours may not. It's up to each of us.