Generally the Catholic philosophical position has been we can know the existence of God by the light of natural reason. By the existence of God, this means we know God exists, but we do not necessarily know what God is, or what his relationship to us is (asides from being the first cause and creator of all).
There is also the saying from the medieval period that there are two books with which we can understand God; the book of nature and revealed scripture. Christ is God’s supreme self-revelation however, and what shows us what God is.
The medievals, based on Augustine and Pseudo-Dionysius, worked out a fairly elaborate theory of ‘the analogy of Being.’ Everything in the cosmos was seen as God’s goodness, life and love coming into creation and the things that are made, all good, true and beautiful. The medievals had a very deep appreciation for the beauty of the material world, as can be seen in the Franciscan saints such as Bonaventure and St Francis and also in the great Dominicans like Albert and Thomas, who were keen to use the natural philosophy of Aristotle to understand the creation and God’s work in it.
Scripture however was necessary after the fall for humankind to understand God, given our intellects were darkened by sin and the likeness to God lost. In the incarnation God glorified human nature and through baptism and the sacraments and the religious life, along with the pursuit of philosophy, humanity could be restored to God. It was seen by many medievals that the existence of God was self-evident, and a number of arguments were given to prove God’s existence.
However, with the rise of modern skeptical philosophy, these proofs from creation were challenged, as was the authority of revelation and religion, especially during the Enlightenment until now.
I think Catholic philosophers need to recover the sense of the good, true and beautiful in the universe and relate these to God’s Being, but this will require new arguments and approaches in light of science especially. Hopefully another Aquinas will be born, soon.
Many people don’t accept God’s existence as self-evident, though I think it is reasonable to answer this by arguing it is rational to believe God exists.