It is not until page 6 that Feser so much as waves his hand in the direction of an argument, when he recounts what he claims was his own voyage of philosophical discovery. He cites Frege as motivating his Platonism. This naturally makes me wonder what Feser thinks about Russell's Paradox, and more importantly of course Gödel's theorem. But he never talks about such subjects, nor does he talk about subjects like the axiom of choice or Euclidean vs. Non-Euclidean geometries. It makes it hard for me to take seriously his analogy between mathematical and philosophical knowledge when he seems to have such a poorly developed theory of mathematical knowledge.
He also mentions that Russell influenced him by showing how little we can know of the intrinsic nature of the material world. Of course, I think Russell was right about this, and in fact that he didn't go far enough. Feser draws a contrary conclusion, perhaps on the basis that knowledge of intrinsic matters is the only real knowledge or that all knowledge must trace back to knowledge of intrinsic matters. But he doesn't prove or argue for either of these claims, at least not here. They have been much discussed in contemporary philosophy, so this represents another area where he is wrong to say that his really important issues have been ignored by the modern naturalists.
He also describes Richard Swinburne as someone who employs "the most rigorous of modern philosophical methods to the defense of religious belief." I am skeptical of this as a description of Plantinga (who also gets this praise), but applied to Swinburne, this can only be considered laughable.
Feser claims that to the naturalists, natural selection is a "pseudo-deity." I suppose it has features in common with how Feser takes God to be, in that it is knowable a priori. Once you understand natural selection, it is quite obvious that it must happen in any situation where there is a mix of some stability over time and some more or less random change. Of course, that natural selection is responsible for specific phenomena, e.g. the diversity of life, requires empirical evidence in each case (evidence which is readily available in the case of many biological phenomena). However, Feser's claim that natural selection could not in any "true or interesting sense" manifest design is unargued. I suppose it depends rather heavily on what one considers interesting.
The remainder of the first chapter continues to provide no arguments, except a sort of inductive argument based on cases for Feser's theory of what motivates secularists. Not all (only most) of his theories about this are wrong, though it goes without saying that they are all absurdly charicatured. The naturalistic world view rejects ultimate authority. That's what it is to be a naturalist. Some, perhaps most, naturalists inconsistently treat naturalism itself as an ultimate authority, because people have trouble with the idea that there really is none. They deserve to be called superstitious, though the fact that naturalism is so purely negative, consisting of little more than the rejection of all ultimate authorities, makes taking it as an ultimate authority a less bad error than most other cases of belief in ultimate authority.
However, contra Feser, there are good reasons to reject ultimate authority. The concept is incoherent. This is admitted by some of its defenders (e.g. Kierkegaard, or Heidegger), who insist that we must believe in it despite its incoherence; this is the reason that "faith" has become the popular line among defenders of ultimate authority. Feser, of course, has no patience with this line, and indeed doesn't even bother to mention why he thinks so many on his side seem to welcome putting things in terms of faith. It might be an enlightening topic for him to investigate.
Feser is obviously right to note that most atheist philosophers do not confront the issue of the existence of his God directly, instead engaging in various smaller detail projects in pursuit of naturalism. But this is because the absurdity of his God as an ultimate authority is widely recognized, while forms of ultimate authority in narrow areas are less obviously unacceptable and so more controversial. This is one of the areas where I agree with Feser; I think the reasons for rejecting his God are closely related to reasons for rejecting non-Humean causation, various forms of anti-reductionism, Armstrong-style universals, perfectly natural properties, and many other still popular philosophical speculations. Still, the connection is not so strong that anyone who rejects God is obviously rationally required to reject all these more moderate views. Feser naturally wishes to run the inference in the opposite direction, and argue that anyone committed to any of the more moderate views, and most naturalistic philosophers are committed to some of them, are committed to his God; the inference is also not immediate in that direction, as I mentioned in my review of his book on Amazon.
Feser ends his first chapter by asking atheists like myself to consider the possibility that we might be wrong. I do frequently consider the possibility that I am wrong to reject the God of Plato, Spinoza, and Einstein. Though that has little to do with Feser's God, I'm afraid it's the best I can do.