(1) The rational foundations of belief in God. We believe things, according to shepherd on the basis of "what we conceive to be the consistent relations of ideas present in the mind" (EPEU 151). We saw this all through the discussion of the foundations of our belief that there is an external world. However, closely analogous reasoning works for the foundations of our belief that there is a God, too. As Shepherd puts it:
For after some contemplation upon the phenomena of nature, we conclude, that in order to account for the facts we perceive, "there must needs be" one continuous existence, one uininterrupted essentially existing cause, one intelligent being, "ever ready to appear" as the renovating power for all the dependant effects, all the secondary causes beneath our view. (EPEU 151-152)
In other words, we are again dealing with a cause that is continual, external, and independent. This line of reasoning can, in fact, go several different ways depending on what, precisely, is the effect with which one begins. And Shepherd doesn't herself draw out the reasoning at any great length. It is clear, however, that she recognizes at least two kinds of arguments for God's existence, a cosmological argument and a modified design argument. The cosmological argument:
Whatever variety and changes of beings there are, all changes must finally be pushed back to that essence who began not, and in whom all dependant beings originally resided, and were put forth as out goings of himself in all those varieties of attitudes which his wisdom and benevolence thought fit. (EPEU 189)
This argument she seems to regard as demonstrative. She also accepts an argument involving final causes, which, she says, "is an argument, though short of demonstration, yet of the highest analogical proof; and one which determines our conduct in human affairs invariably and irresistibly" (EPEU 353):
Amidst the apparent contrivances which mortal beings have had no hand in arranging, it appears impossible to descry, or detect, the point where mind perceived possible qualities, and directed the aptitudes of various motions, but that mind must be the cause of that which the understanding concludes to be contrivance.... (EPEU 353)
Elsewhere she also gives an argument that may or may not be intended to be the same argument; she seems to regard it as a stronger argument, but it's possible that the argument is taken as stronger not as an argument for God's existence but as an argument that God must be intelligent:
...since we perceive instruments in existence which are means to ends, there must be the director of motion, the perceiver of ends, the former of instruments int he universe;--perception of ends and direction of means, are mental qualities; are the properties of teh continued existence, called mind; mind therefore must have been at the fountain head of these contrivances; but not a mind whose existence is more invisible than that of our own minds to each other.... (EPEU 389-390)
(2) The knowledge of our own independent existence. This is what is often known as the problem of personal identity, and as Shepherd notes, it is really in great measure a problem of how we can know that we continue to exist even independently of our own perceptions. The cause of sensation in general cannot logically be the same as the causes of particular sensations; we have actual sensations only when causes of particular sensations interact/unite/mix with the cause of sensation in general. Every sensation is felt to be a beginning-to-be, that an effect, arising from the union of the ability to sense with the ability to make itself sensed. Using reasoning similar to some of the reasoning we used in looking at the external world, we can draw the conclusion that the readiness of our capacity for sensation in general -- that is, our mind -- to appear requires it to continually exist even when sensation is interrupted, as when we are deeply asleep. This is confirmed by our sense of continual existence, which is the combination of memory with sensation.
(3) The distinction of mind and body. The distinction of mind and body actually falls out immediately from Shepherd's account of the external world: the mind is the capacity for sensation in general and it is simply distinct from body, which is "the continually exciting cause, for the exhibition of the perception fo extension and solidity on the mind in particular" (EPEU 155). These can't be the same thing without making it impossible to appeal to them to explain our sensations.
When people talk about the distinction of mind and body, however, they are usually interested in whether mind and body are separable, that is, whether the mind can continue to exist independently of the body. Here, interestingly enough, Shepherd says we can't say for sure. There is nothing in our experience that guarantees that they are not separable; but she thinks we don't know. She herself does, however, think that the resurrection of the dead, "or at least an existence analogous to it" (EPEU 157), is possible and probable; arguments against its possibility fail and, as we might say, it makes sense of the relation between mind and body. In talking about the resurrection, Shepherd allows herself an unusual bit of flowery language, comparing the human person to a butterfly:
It would appear therefore equally inconclusive for man to argue against the possibility of a future life on account of the dispersion of the particles of the present gross body by death, as for the worm to suppose it could not again live because its outside crust wholly perishes:--He might resist every notion (however prompted by his instinct or his wishes,) of an existence beyond the range of his present experience, beyond the extent of the leaf on which he is born to die; yet the time would equally arrive, when as a winged insect he would roam through boundless space in comparison of the circumscribed spot to which his former existence was confined, and chase the brilliant image of himself, through a live-long summer's day, amidst the sweets of a thousand flowers. (EPEU 158-159)
(4) Instincts and prophetic visions. By 'instincts' she means what we usually mean, e.g., "the instincts of birds give them notions of the materials requisite for making their nest previously to their first formation" (EPEU 160). The existence of instincts is simply manifested by the kinds of effects we see, just as with the external world or with dreams; the only difference in question being the kind of effect in question. The same thing, however, is the case with prophecy: whether anyone has prophetic visions is simply a matter of whether there are any effects requiring such a cause.
(5) The natures of unperceived objects. Since Shepherd is quite insistent that we have no direct knowledge of the external causes of our sensation, that they are in some sense unknown, one might conclude that we simply have to be agnostic about everything actually in the external world, insofar as it is unperceived. Shepherd actually thinks, however, that we can have an indirect knowledge of some of the characteristics of these unperceived existences. We get this by taking what we do perceive and subtracting the qualities that all sensations share. Remember, sensation is the union or mixing of the capacity for sensation in general (mind) with particular causes of sensation (things in the world); thus what sensations all share is derived from the mind. Comparing our sensations, we can recognize these features and abstract from them in order to get a sort of barebones outline of the things themselves. We've actually already seen this: Shepherd thinks that Berkeley's argument that there are other minds is a good argument and that we can get the conclusion that there are other things in the external world besides minds by using a similar argument but subtracting all the mental aspects. Further, variety in the effect indicates some kind of variety in the cause, so we can draw conclusions in this way as well.
There are probably other topics that could be added from other places in Shepherd's works, but these are the ones she explicitly addresses. I think she is arguably doing more than might first appear here. This isn't just a brief tour of ramifications, but an argument in its own right, simply one that is given indirectly. There are a number of views one might have about how we end up accepting that the external world continues to exist independently of our mind. One could hold that it is a mere assumption, perhaps a natural belief, not grounded on anything further (this is the position of the Scottish common sense philosophers). One could also hold that in fact this is a trick of the mind, and that we get this belief by confusion (this is the position of Hume, although he supplements it with an account in which it is a hypothesis that we confirm). Or one could hold that we accept it on fully rational grounds. Only the third kind of approach, which is the kind Shepherd champions, allows us to say that we know that there is an external world. And the causal account Shepherd bases her arguments on is a fairly minimal one. But here she is making a broader point: Any account of the external world strong enough to allow us knowledge that it exists is an account that also makes it possible to use similar reasoning for cosmological and teleological arguments for God's existence, for personal identity, for the failure of arguments against life after death, for prophetic visions, for scientific realism, and so forth. I say 'possible' because whether we can actually reasonably do so simply depends on what effects we actually experience; any approach to the external world strong enough to give us knowledge of it (rather than just an assumption or leap of faith or trick of the mind) makes it impossible to rule out the possibility that we could know such things, if only the effects we experience are of the right kind. As I said at the beginning, one's account of our knowledge of the external world has effects all over our philosophical view.
(I should say as a side note that Shepherd is very, very copious in her use of italics; for purposes here, where so many italics might distract, I have simply ignored her italics when quoting her.)