I've summarized the two sections that comprise the two halves of the critique of traditional religion contained in Hume's First Enquiry here and here. Now I want to take a look at how the two arguments complement each other and evaluate how good a critique they make.
Although he is not explicit in this, I take it that Sections X and XI of the First Enquiry are intended by Hume to jointly show that neither the miraculous events of revealed religion nor the conclusions of natural theology can serve the purpose of acting as a rational foundation for religious belief. Hume leaves it to the reader to evaluate whether religious belief is rational without these supports, but the implication is pretty clear that he thinks it is not.
As I see it, there are three main approaches to establishing the foundations of religious belief in the history of philosophical theology. First, there is the a priori approach, exemplified by the sorts of arguments that occupied mediaeval theologians. The various formulations of the ontological argument, the first cause argument, etc. are ways of attempting to establish the existence of God and determine his properties through an analysis of the contents of our concepts and by applying deductive logic once those concepts are clarified. Second, there is revealed religion, which calls on miracle accounts, authority, tradition, scripture, and revelation to provide the epistemic justification for religious belief. Third, there are a posteriori approaches to religion exemplified in natural theology. This approach looks at the way the world is and argues that the existence of God is a necessary component of any adequate explanation of the world.
Hume's critique is targeted at the second and third approaches. His critique of revealed religion is primarily occupied with the lack of foundation of miracle accounts, but the argument is easily extended to all forms of justification of revealed religion. There is no logically sound reason why we should believe anything on the basis of authority, scripture, tradition, or revelation. There is simply no logical connection between the speaker of a proposition and its truth value.
Our confidence in each of these bases must then rely in their perceived reliability. But experience is the only way to determine what is reliable, and so the content of a claim made in scripture or from the pulpit has to be judged according to the likelihood criteria by which all other claims about matters of fact are judged. And in the case of vast swaths of scripture, tradition, etc. the overwhelming evidence is that the way religion describes how the world is (with miracles, angels, talking donkeys, resurrections, etc.) is far different than our normal sensory or scientific experience of the world. This being the case, we ought to apportion our beliefs to the evidence and reject revealed religion.
Hume's critique here is devastating to the method of justification of revealed religion. An advocate of revealed religion will have to offer a plausible reason to think that authority, etc. are generally trustworthy that does not call on reliability, because as Hume has shown, the probability (and thus general reliability) of scientific accounts of the world that contradict religion is far greater than the probability that the miraculous events that pepper scripture and tradition actually occurred. This reason won't be able to point out a necessary connection between the source of information and its truth value because no such connection exists. And it won't be able to claim that we can trust revelation, authority, or scripture as true or miracle accounts as likely because of the truthfulness or power of the deity, because the existence of this deity or the truth of religion is supposedly justified by these things. Invoking God to buttress the veracity of the scripture (or authority, etc.) which is used as a reason for positing God's existence is as straightforward a circularity as you could ask for.
The reach of his critique of the Design Argument is much more limited, however. A posteriori proofs for the existence of God are essentially inferences to the best explanation, and at the point at which Hume was writing in the mid 18th C., there was not much in the way of scientific theory that could conclusively argue against a divine creator being a candidate for the best explanation for the existence of the universe and the order that we see in it. Newton's universe was a static place with fixed laws and no discernible origin; it is not until the development of modern cosmological theory that anything approaching an understanding of the beginnings of the universe was developed. Similarly, Darwin's extraordinary achievement of detailing a natural mechanism to explain adaptation and order in the biological world was still a century away. So, Hume was unable to fully refute the Design Argument's claim to have established the likelihood of a deity.
What Hume's argument did accomplish, though, is still significant. By correctly identifying the Design Argument as an argument from effect to cause he was able to show just how little advocates of that argument can hope to get from it. All it warrants is the inference to some being or entity of we know not what sort with we know not what properties that is sufficient somehow to bring about the existence of the universe in some fashion. Not exactly the anthropomorphic God of traditional religion. Nothing more can be concluded because for the other attributes traditionally attributed to God (omniscience, omnipotence, goodness) there is scant evidence in the world we see. The Design Argument gives us none of these properties. And, having blocked the methods of revealed religion, Hume has also blocked one avenue of supplementing the minimal conclusions of the Design Argument to get something approximating Christian dogma.
Hume's critique does not address the a priori method of establishing the existence of God, however. My supposition is that his Empiricist worldview would have led him to simply dismiss any a priori arguments attempting to establish the existence of God, or for that matter, any matter of fact. He probably was right to do so, but defending that position is a task for another day.
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