I have been a big fan of Greek mythology as long as I can remember, but my first exposure to Greek mythology in literature, like most people, was the Percy Jackson series. I love pretty much anything to do with Greek mythology, whether it be games, or mentions of the gods in stories and film/TV, but I'm never certain where to look for accurate stories about the deities of Olympus (or the underworld) due to oral tradition spreading misinformation which then gets documented down in literature.
If anyone knows original stories to read to get an accurate understanding of these Greek Tales, it would be greatly appreciated.
PS. If anyone also plays games, can you tell me the accuracy of zagreus' story in the game Hades?
Authentic, or a good read? There are popular accounts out that there that are both accurate and a good read, to be sure, like Stephen Fry's books Mythos, Heroes, and Troy. There are disadvantages: in a streamlined narrative like that you never get much sense of how much variation there is in Greek myth, and you also don't get alerted to the fact that some of the stories are very old, while others are very late, and that often we can see what context they arose in and sometimes different bits of a story come from very different contexts. Also, I believe Fry's books don't have indices.
(I urgently advise you NOT to go for Robert Graves, by the way. He is not accurate, and he routinely makes up large swathes of the stories, in a much less transparent way than in Fry.)
If you're looking for a reference work, then a good myth encyclopaedia is what you need. And there's one correct choice. For a general interest reader, the clear best of the bunch, by a very long way, is Jenny March's Dictionary of classical mythology. Her book is a sensible size, doesn't skimp on details, is very very accurate, and always tells you where to find the stories in ancient sources. Don't settle. If you're doing full-on historical research then there are better choices, but from what you say it's clear that March is the one you should get.
As for Zagreus: spoilers for the game follow. /r/AskHistorians doesn't have rules about spoilers at the time of writing, so if mods judge that my use of spoiler mark-up isn't appropriate, let me know and I'll remove it.
The game doesn't stick closely to his story, and it certainly whitewashes the story of Hades himself. I'll focus on Zagreus. In ancient sources Zagreus isn't the son of Hades, but he is the son of >!Persephone!<. Also, in >!Orphic religion!< he was indeed treated as >!the same person as Dionysus, and torn apart by the Titans!<. Just to advertise March's book, here's her entry:
Zagreus. A Greek god, >!identified with DIONYSUS,!< who was worshipped by those who practised >!the mystery religion of Orphism!<. Their story of his birth is the only significant divergence from the usual myth. ZEUS, they said, lay with >!PERSEPHONE!< in the form of a snake, and from their union Zagreus was born. Zeus intended this son to be his heir, but his jealous wife HERA >!persuaded the TITANS to kill the child!<. They distracted his attention with toys, then carried him off, >!tore him to pieces and devoured him, all but the heart,!< which Athena rescued and gave to Zeus. >!From this heart, still beating,!< Zeus >!refashioned!< his son in the body of his mortal love SEMELE. In due course the child was born a second time, though not without further intervention by the jealous Hera.
Zeus blasted the Titans to ashes with his thunderbolt, and from this residue was created mankind. This explained the mixture of good and evil in men, for they incorporated a tiny trace of divinity in a great deal of Titanic wickedness. The >!Orphics!< believed in the transmigration of souls, and in the possibility of stamping out man's inborn evil during three virtuous lives and intervening periods of purification in the Underworld. After this the believer would dwell in Elysium for ever.
[Diodorus Siculus 3.62, 3.64; Pausanias 7.18.4, 8.37.5; Nonnus, Dionysiaca 5.563-71, 6.155-205; Ovid, Metamorphoses 6.114.]