I'm an aspiring film-maker and I would like to read actual Greek Myths in order to make a film, yet I can't seem to find an actual place to read these Myths, just summaries. So if anyone has a website where I can actually read the original myths I'd appreciate it.
P.S. No, greekmythology.com does not have the actual myths
That text doesn't exist because it can't exist. Ancient folk traditions (like all folk traditions) were fluid and often contradictory. Some belief systems become codified into a “set” written tradition, but there is rarely a dogmatically accepted “true and accepted” official/original version.
Even Christianity, which prides itself on having a “set” written version of its story was faced with an early fluidity. The fourth-century Council of Nicaea and the longer process of discarding various books that had been written began to codify a “true version” of Christianity and the New Testament, but that was only after years following most of the events described. Before that – even though Christianity has consistently looked to written texts – there were many versions.
For the ancient Greeks, there was no Council of Nicaea. In addition, there were many centuries of writers who addressed their traditions in one way or another. Throughout this literate period and across the Greek-speaking world, variation was key. As one should expect with all folklore.
Others perhaps can suggest the best sources to look at, but keep in mind that whatever you read is only “a” version, not “the” version.
Edit: I have edited this to remove the "flash" points of the Council of Nicaea. My main point here is that there were many interpretations and various stories floating around - in print and perhaps more circulating orally - that early Christians had to deal with to arrive at an "official" version. Nicaea's role was admittedly narrow. Thanks to those who commented; not sure why some comments have been removed.
/u/itsallfolklore is absolutely right to emphasise the oral origins of myths. I want to go into how we get from those oral origins to the kind of thing you see in modern retellings.
Greek myths, as compiled in modern retellings or encyclopaedias, are derived from a variety of types of evidence that we have access to:
1. Pictorial. We've got heaps of paintings and statues from ancient Greece, and many of them are closely related to stories that exist in textual versions. Some, however, appear to have been primarily in pictorial form, including a couple of famous ones: (1) the story of Achilleus chasing down and murdering the young prince Troilos; (2) the story of the Gigantomachy, where Herakles rescues the gods by fighting the Giants (a version of which fed into the plot of the Disney Hercules). We basically don't have textual versions of these stories, apart from some very late retellings.
2. Poetic. I suspect this is probably the kind of thing you had in mind in asking your question: poetic accounts of familiar myths and legends are often popularly seen as 'original' versions of the stories. Very approximately in order of the age of the poems, they include cosmogonic poetry (origin stories about how the 'present' divine order got into place); heroic epic, especially Homer (stories about famous legendary wars, particularly the Trojan War); Athenian drama, which touches on all kinds of heroic legends (a wide range of stories of heroes of the past, not so much about the gods); and later literary retellings of older material (such as Apollonios' Argonautika, about Jason and the Argonauts).
The trick is that all of these, including the very oldest ones, are reworkings of pre-existing material that we don't have access to. It's most obvious in the case of something like the Argonautika, where we know older poetic versions did exist, but haven't survived to the present day. But even with things like Hesiod, it's pretty easy to read between the lines and see that the poem is a reworking. There's no such thing as a pristine original; there's just retellings, some older, some later.
3. Mythographic. From the 3rd century BCE onwards, people liked having stories in an accessible form. Prose retellings became fashionable, both for scholars and general readers. The result was retellings and encyclopaedias that look quite similar to modern ones (but generally not as well organised!). Some key texts are the Library, or Library of Greek mythology (also known as pseudo-Apollodoros), and the Fabulae (pseudo-Hyginus). These may not sound very 'original', but in many cases they are the only kind of evidence we have.
4. Other references in passing. Many ancient sources refer to stories in passing that are not known by any of the above avenues. These, too, don't sound very 'original', but for some myths they're all we have.
There's a couple of things worth remembering in looking at this material. First, as I said in reference to the second category, the versions that seem 'original' to some modern observers were already retellings, reworkings, or retcons.
Second, modern observers have a ridiculously strong bias towards textual sources. Take a look at the Wikipedia article on Troilos: the first thing it does is spend a huge amount of space on looking at available textual sources. Only after finding that they're not very substantial does it turn to the pictorial sources, which are the main avenue by which we know the story. There's a strong sense in this article that only a textual version of the myth is a 'proper' version. The whole of the first two paragraphs, which are ostensibly based on the earliest textual source, are in fact entirely derived from a two-word allusion in a prose summary of an early poem. We can be very confident that the poem did not treat the story in any kind of depth. The most important medium for the story was pictorial all along; but the author of the article just cannot bring themselves to see it that way. Scholars of early poetry are prone to the same bias, sometimes insisting that any allusion to old material must be alluding to a poem.
So this is where /u/itsallfolklore's answer comes in. The oral material that lies behind all of these categories of evidence is entirely lost to us, but in many cases must have been the true 'origin' point for a given story.
There are probably many exceptions. The stories of Troilos and the Gigantomachy are so strongly biased towards pictorial sources that that may well be how they actually originated: as pictures, not verbal stories. Some stories originated in literary versions: for example, the story that Medusa's transformation was a 'punishment' for being a rape victim doesn't appear in any early sources -- it may well have been invented by the Roman poet Ovid, in his poem the Metamorphoses.
To get access to the oldest available material that we do have, I recommend a combination of two books: (1) Jenny March's Dictionary of classical mythology, which cites ancient sources for all the stories that she includes; that will enable you to go and look at the relevant ancient material in each case. (2) One of the ancient myth encyclopaedias: the pseudo-Apollodoros Library is the obvious candidate, and it exists in many modern translations. As it happens there's one edition, a book called Apollodorus' Library and Hyginus' Fabulae, translated by R. Scott Smith and Stephen M. Trzaskoma, which includes two of these encyclopaedias for the price of one.
Finally, you may have noticed I've said nothing about religion. Myth has a very indirect relationship with religion. Some stories do appear to be closely related to religious cult at one particular holy place or another, but the sources that we have generally are not. There are exceptions. Two of the Homeric hymns, the Hymn to Apollo and the Hymn to Demeter, do appear to have had a close relationship with religious observances at Delphi, Delos, and Eleusis. But for the most part, the stories that ancient sources tell us are many steps removed from actual religious practice. They have no more to do with ancient religion than Milton's Paradise lost has to do with stuff that actually happens in a Christian church. For a deeper perspective on the relationship between religious practice and the surviving evidence about mythical stories, I recommend Ken Dowden's book The uses of Greek mythology (which is available for free on the Internet Archive).
While it is very interesting reading about the oral tradition of Greek myth and how we may never know the true origin of myths because of that, I feel that that isn't really getting to the heart of your question. So let me offer you a concrete solution.
I linked Hesiod's Theogony, which is one of the best starting points for Greek mythology since it starts at the very beginning of their cosmology. Theoi includes translations of most of the ancient sources for Greek (and some Greco-Roman) myths, as well as a nifty little feature that lets you look up gods/monsters/heroes and the page will have all the passages related to their myths cited.