What does the Greco-Roman religion believe?
Greek religion, spreading as it did over many centuries and many different city-states, incorporated a great deal of variety in their beliefs. Nevertheless, the "pantheons current among different communities have enough in common to be seen as essentially one system, and were generally understood as such by the Greeks." 
It may be assumed that Greco-Roman religion is an antiquated belief system; however, this is only partially true since it has not only influenced other religions in the world today, but elements of Greco-Roman religion are still present today in other worldviews. (See Greco-Roman timeline)
That Greek religion was polytheistic is clear, but it also incorporated concepts that could be said to resemble an Ultimate Reality. Even Zeus, the mightiest of all gods, was subject to the powerful force of Destiny or Fate. The Delphic Oracle told Lyidan inquirers that "no one, not even the god, can escape his appointed fate." 
At the same time, however, the Olympians regularly directed the fate of human beings and one of Zeus' many epithets was Moiragetes, "guide of fate." Fate, while not a personal god, was nevertheless "half-personal because so clearly moralistic." 
In Greek philosophy, this concept of a Supreme Law or Ulimate Reality was much more emphasized, often at the expense of traditional beliefs about the gods.
The ancient Greeks viewed the earth as a flat disk floating on the river of Ocean. In Plato's Timaeus, the world is treated as a living thing, with body and soul.
The primary source for the ancient Greek creation myth is Hesiod's Theogony. According to this account, four divine beings first came into existence: Chaos, the Abyss, Earth (Gaea) and Love (Eros). Then the world came into existence when Earth was forcibly separated from her consort Heaven (Uranus) for a time so that she might give birth. To effect this separation, Uranus' genitals were severed by his son Cronus (the father of Zeus) and thrown into the sea, from which rose Aphrodite. 
Spirits, Monsters and Other Mythological Beings
Another aspect of Greek religion worth mentioning is the set of mythological and sometimes monstrous creatures that populate its myths, the most notable being the following:
- Amazons - race of female warriors
- Keres - evil female spirits
- Medusa –a winged female monster with hair made of snakes.
- Satyrs – half-man, half-goat nature spirits who dwelled in woods and mountains and were lusty followers of Dionysus
- Centaurs – half-man, half-horse creatures who were wild and lawless but possessed cultural knowledge
- Sirens -
- Typhon - represents disorder and devastation
Human Nature and Purpose of Life
Plato emphasized the existence of a soul that is separate and distinct from the body. He also insisted on its natural immortality.
Religiously speaking, the most important thing to do in life is believe in the gods and perform the proper sacrifices and rituals. This would avoid reprisals both from gods and fellow human beings and encourage gifts from the gods. Greek religion was this-world oriented; any postmortem benefits of religious beliefs and actions were only peripherally considered, if at all.
Death and the Afterlife
"I'd rather be a day-laborer on earth working for a man of little property than lord of all the hosts of the dead." --Achilles, in The Iliad
As illustrated by the above remark by the hero Achilles, death was not a glorius thing for the ancient Greeks. In Homer's epics, the dead are "pathetic in their helplessness, inhabiting drafty, echoing halls, deprived of their wits, and flitting purposelessly about uttering batlike noises."  While undesirable when compared with life on earth, this vague, shadowing existence was not generally cause for fear of the afterlife. Only terrible sinners (like Tantalus, Tityus and Sisyphus) were punished after death; similarly, only a select few ended up in the paradisical Elysian Fields.
With the rare exceptions mentioned above, Hades was the universal destination of the dead in Greek religion until the latter half of the 5th century BCE. Hades was a cold, damp and dark realm that was guarded by the god of the same name. The "gates of Hades" were guaded by the fearsome hound Cerberus, who wags his tail for new arrivals but does not allow anyone to leave. Without proper burial, one cannot enter the gates of Hades. The river Styx is the boundary between earth and Hades, but Hades has other rivers as well (e.g. Phlegethon, Acheron, Cocytus). A similar concept is found in Japanese Buddhism in the Sanzu River, which the dead must cross on the way to the afterlife.
In Greek religion, Tartarus was the deepest region of the underworld, lower than Hades. Hesiod wrote that it would take an anvil nine days to fall from heaven to earth and another nine to fall from earth to Tartarus. Hades, not Tartarus, is the place of the dead but some especially wicked characters have been imprisoned in Tartarus to be punished. It is where Sisyphus, thief and murderer, must repeatedly push a boulder up a hill for eternity; where Ixion, who killed his father-in-law, is attached to a flaming wheel; and where Tantalus is kept just out of reach of cool water and grapes for sharing the secrets of the gods with humans. Tartarus is also where monsters and other enemies have been cast after being defeated by the gods, including the Cyclopes, the Titans and Typhus. In Roman mythology, Tartarus was the eternal destination of sinners in general.
Elysium (also called Elysian Fields or Elysian Plain) was a paradise inhabited at first only by the very distinguished, but later by the good. Elysium first appears in Homer's Odyssey as the destination of Menelaus. It is located at the western ends of the earth and is characterized by gentle breezes and an easy life like that of the gods. Closely related to Elysium is Hesiod's Isles of the Blessed, mentioned in his Works and Days, which was located in the western ocean.
The notion that the human soul enters another body upon death, though unfamiliar in popular Greek religion, was widespread in Greek philosophy. The doctrine of transmigration is first associated with the Pythagoreans and Orphics and was later taught by Plato (Phaedo, Republic) and Pindar (Olympian). For the former groups, the soul retained its identity throughout its reincarnations; Plato indicated that souls do not remember their previous experiences. Although Herodotus claims that the Greeks learned this idea from Egypt, most scholars do not believe it came either from Egypt or from India, but developed independently.References
- "Greek religion," The Concise Oxford Dictionary of World Religions, 231.
- "fate." The Concise Oxford Dictionary of World Religions, 213.
- The Oxford Dictionary of Philosophy, 5.
- "Greek mythology." Encyclopaedia Britannica; Hesiod, Theogony 126ff.
- "death, attitudes to." The Concise Oxford Dictionary of World Religions, 150.