You’ve come to the right place. Catholic Answers has a free apologetics curriculum that discusses this in some depth. See here:
Curricula: Apologetics
catholic.com/encyclopedia/curricula/apologetics
According to that curriculum:
"the science of apologetics easily falls into three great divisions:
First, the study of religion in general and the grounds of theistic belief;
second, the study of revealed religion and the grounds of Christian belief;
third, the study of the true Church of Christ and the grounds of Catholic belief."
Your question corresponds closely with the second division.
To provide an overview, regarding the first section, the apologetics curriculum discusses the following grounds for religion in general and theistic belief: first (in this section) we prove that there is a God and what His attributes are, especially that He is Omnipotent, Infinite, the Creator, and that He orders the world through Divine Providence. Second (still in the first section though), we show that it makes sense for man to be religious, by showing that we have an immortal soul that will continue to exist after our death and a free will to choose for or against God.
Now, a few of the divine attributes that Christians believe in are already different from the deities that non-Christian religions profess. For example, it is my understanding that Brahma is not considered an intelligent personal being to whom we can relate as persons. And the Greek god Zeus was not considered infinite or omnipotent. (According to the Illiad, if memory serves, Zeus’s brother Poseidon thought he could defeat Zeus in personal combat.) So we’re already diving into some of the Christian distinctives before we’ve even gotten to the second division, the one about why we should accept the Christian revelation. Let’s get to that.
In the second division, which is the one where I think the heart of your question lies, we discuss the following issues: Revelation, Mystery, Faith, Dogma, Miracles, Prophecy. There is good evidence from philosophy that God wants to communicate with us. Evidence includes His providential care for us and the fact that He has endowed us with a rational intellect capable of perceiving Him and a will capable of choosing Him. Then we go on to show that the Bible is God’s revelation. It has characteristics that indicate its origin in God, such as prophecy. There are rational reasons to believe that its prophets really did perform divine acts, especially the resurrection of Jesus, which has objective historical evidence for it.
Moreover, the mysteries proclaimed by Christianity have persuasive power. The mysteries of the Trinity and the Hypostatic Union and the Eucharist are beyond the power of human explanation. Some people might think that means we shouldn’t believe in them, but in fact it is a small argument in their favor: we should expect some divine things to exceed our mental abilities. This is a consequence of God’s infinity. A God who was completely comprehensible would actually not be God, because the unlimited must exceed the limited. So we should expect some Mysteries in the true religion, and we find these in Christianity.
That’s a small argument in their favor. But another argument in their favor (perhaps a stronger argument) is that our Mysteries are demonstrably compatible with reason. If a religion’s dogmas contradict reason, we should reject that religion. But none of ours do. We should expect God to be beyond our comprehension, but not contrary to our reason. He can be a Trinity, but not a square circle. This is because the Trinity is not a self-contradiction, whereas a square circle is. By examining our doctrines, a person can come to understand that they fall exactly where they ought to between two extremes: they are not so human that we can comprehend them fully, but they are not contrary to reason, for then we should reject them. This is a unique feature of Christianity.
So that’s a taste of some of the ideas from this particular apologetics curriculum, but click the link I gave you and dive in deep! This is one of the coolest studies in all apologetics.