I think the argument should begin here in its most fundamental, elemental parts: everything that moves and has parts is moved by something outside of it. A rock is made of parts. When it moves it is not entirely from its own internal energy. The fact that it moves means that there must be something external to it acting upon it.
That holds equally true for everything in the universe that has parts and moves. However, the universe as a whole, which science knows to be finite in time and space, to have had a definite beginning, before which it did not exist, is itself made of parts and moves.
Therefore there must have been a source of existence and movement, a mover, that existed before the universe, that caused it to come to existence, and to move. Yet this mover pre-exists ultimately all movement, causation, parts, and material reality, and must therefore be without movement, parts, physicality, need of anything other than itself for its own existence, uncreated, and eternal. This then at least is an Unmoved Mover.
The second argument is that everything that exists which shows internal organization, purpose, has in its own nature a design. This is seen in the very pattern of the universe, chemistry, laws of physics, the existence of complex bio chemicals especially DNA which encode the design for all living organisms.
Since the universe was caused by the Unmoved Mover, and is inherently based on data, information, rationality, and design, therefore the Unmoved Mover must be an intelligent designer, and since moral choice exists in nature among animals, with free will among humans, then the intelligent designer must have personality and free will to create personalities and free will.
Third, a personal God is manifested universally in culture and among humans since time immemorial, and through personal experience, and is most proven through miracles. From the established fact of an Unmoved Mover, an Intelligent Designer, it is at least plausible that this entity is also personal, and involved in our personal lives. But Catholic miracles, in particular modern day Catholic miracles established as such, demonstrate the truth of Catholicism, and therefore Christ, and therefore a personal God that corresponds to the God of the Bible.
First, is the miraculous image of Our Lady of Guadalupe. After 500 years, the cloth should have decomposed but it hasn’t, there should be paint or some other substance on the cloth corresponding to the image but there isn’t, and there should be no image inside the pupil of the Virgin Mary but there is, at the microscopic level showing Juan Diego with the Bishop of Guadalupe the day he showed him the image. There are many books that describe this miracle and all the particular miracles associated with it. It proves that the Catholic idea of God, ie the Trinity establishing the Catholic Church as the authority, is true, because it shows God sent Mary to appear in Mexico to preach Catholic Christianity.
Second, the Miracle of the Sun at Fatima, where the three children said Mary appeared to them, was witnessed by thousands, on the day predicted by Mary, resulting in tens of thousands of people whose clothes were soaked by the rain to have their clothes instantly dry once the Sun returned to its place in the sky.
Third, the countless miraculous healing at the shrine to Our Lady in Lourdes. Many credible witnesses such as doctors witnessed things like a pilgrim going to Lourdes missing a leg, to have the leg literally grow back right in front of the witnesses’ eyes.
Fourth, modern day Eucharistic miracles showing the true, physical preserved of Christ in the Eucharist. Hosts that turn to flesh, when examined under microscope are heart tissue. Blood from hosts verified to be human and male. Certain occurrences observed carefully by scientists to exclude fraud. There are many such documented miracles just in the past century.
There are many more Catholic miracles just in modern times, not to mention the countless miracles throughout Church history. If you arrive at a personal God whose miracles verify Catholicism, then it takes less science and rationality, and more simple faith at that point, to accept in total the miracles of the Bible, especially those by Christ that demonstrate His credibility and divinity, especially the Resurrection, and all the prophesies of the Old Testament His life miraculously fulfilled, also demonstrating His divinity.
Therefore, any honest atheist or agnostic can arrive at the certainty of God’s existence and the Catholic Faith. In addition to the many other more secondary reasons to believe such as the Church establishing Western civilization, the beauty of Catholic culture, and the social and economic principles of the Church that help ensure a just and orderly society. At some point, considering the mountain of evidence, the unbeliever has to ask why not believe? How much more evidence do you need? Those who are honest and humble will recognize the biases of our anti-Christian culture ingrained into young people, indoctrinating them against theism and the God of the Bible, with secularism and atheism.
In conclusion, God exists. He is personal and corresponds to the God of the Bible. He proves this through miracles, especially to modern people by means of modern day miracles, once verified leading to a total acceptance of the Bible, Christ, and the Catholic Church.
But once that sets in, it is a terrible and life-altering truth, ie the Catholic, Christian Faith, since the believer must accept the reality that this life is a valley of tears, a test for our salvation, not meant for pleasure and comfort, very possibly a time of great penance, after which most will either go to hell or the fires of purgatory. That it is better to suffer extreme hardship, illness, and poverty now, than the punishments of purgatory.
For the honest atheist who already knows what the Church teaches, that is probably why they resist theist arguments and conversion in the first place, because they know what the consequences are, to give up sin, do penance, to detach from this life, and to completely follow Christ.