"Religion is based on faith, and faith means pretending to know what you don't know. Faith means not asking questions, not thinking."
Religion is certainly based on faith, but not on blind faith. There are plenty of rational arguments for the existence of God and the truth of the Catholic faith.
“Can the existence of God be proven?”
Yes. The Catholic Church teaches that we can be certain of the existence of God, through the use of our reason. There are many philosophical arguments for God’s existence. The most famous are the Five Ways of St. Thomas Aquinas. These are detailed arguments, very often caricatured and simplified by atheist writers. They need to be explained in detail and cannot be presented adequately here.
“Can all Catholic doctrine be proven?”
No, the Church does not claim that it can. Some of the things the Church teaches have to be taken on faith. Again, this isn’t blind faith. There is evidence for all Catholic teaching, but—unlike the existence of God—the Church does not claim that we can know the truth of everything it teaches just by using our reason. Faith is involved.
“Surely faith is irrational?”
Not at all. Our day-to-day lives would be impossible without faith. We have (to some extent, at least) faith in doctors, faith in government officials, faith in the bank, faith in the news reports that come to us from around the world, faith in the airplane that takes us thousands of feet in the air. Our faith in these things is not blind or irrational, but based on experience and judgement. The same goes for the Catholic faith.
“Extraordinary claims require extraordinary evidence, don’t they?”
Absolutely, and the evidence the Catholic Church presents is extraordinary. It can be argued that the Catholic Church is the oldest continuing institution in the whole world. In a history of more than two thousand years, it has never changed its essential teachings, even in the face of many persecutions. It was spread by an obscure group of preachers—Jesus Christ’s apostles (followers)—who were willing to be be killed (and most of whom were killed) for telling the world that Jesus had died and risen from the dead. If this was a lie, they would have been extraordinarily stupid to stick to it even in the face of persecution and death.
There are many other extraordinary evidences for the truth of Catholicism, including the lives of the saints, well-documented miracles (in modern as well as historical times), and the fact that the Catholic faith flourishes in every part of the world. Many books could be written on this subject. Indeed, many books have been written on this subject.
“I believe in some kind of Supreme Being, but not an old man with a white beard sitting on a cloud.”