I put the words in quotes cause there’s generally no definite start date to any religion, and there are modern practitioners of just about any religion to this day. I guess if I were to expand my question out, it would be “what is the earliest evidence of Greek paganism and/or the estimated dating of its early practice; and when did both Greek and Roman paganism become minor world religions, completely overtaken by other religions, especially Christianity?”
"paganism"—a late antique term referring to any practice that was not specifically Christian. A pāgānus in Latin is simply a country dweller, a peasant, a yokel. It is not a useful term here, because it is meaningless before the 5th century CE.
"Greek" religion—the common collection of practices generally shared by those speaking a dialect of the Greek language and including shared mythology, ritual practice, sacred architecture, etc.
"Roman" religion—the common collection of practices generally shared by the early Iron Age culture which developed on the plains of Latium in central Italia, including (again) a shared mythology, ritual practice, sacred architecture, etc.
I am only writing about the "beginning" of these religions.
There is evidence of "Greek" religion at Olympia all the way back into the 10th century BCE, in the depths of what used to be called the "Greek Dark Ages" and only a handful of decades after the final spasms of the Bronze Age collapse. This from excavation of Olympia's ash mound, where the priests of Zeus would immolate animals in the ritual exchange detailed by the Prometheus myth. By the 8th century BCE, we have good evidence of prototypical Greek temple architecture (the terra cotta temple models at the Sanctuary of Hera at Perachora, e.g.).
But before the "Greeks" were the Bronze Age Mycenaeans, and though it is very difficult to get a sense of Mycenaean "religion," it is now generally accepted that Mycenaean practice colored and shaped later Greek religion. Many of the familiar gods of the Greek pantheon are named in Mycenaean Linear-B texts, including (often) Poseidon, Artemis, Athena, Dionysus, cow-eyed Hera, and Demeter. Mycenaean cult, in turn, was colored and shaped by Minoan practice. Demeter in particular is almost certainly not Greek at all, but *pre-*Greek, belonging to that obscure epoch before the arrival of the Indo-Europeans into Europe in the time before history.
As you can see, it is very difficult to say when Greek religion "began." In its Classical form, the distinctive elements which seem to set it apart from Mycenaean religion are coming together in the aftermath of the collapse, about 1050 BCE and more or less fully realized by 750 BCE.
Likewise, it is difficult to say when "Roman" religion begins. It shares many elements with "Greek" religion because the early Romans were in regular contact with Greek neighbors on their southern borders, in the Bay of Naples. The fundamental elements of the two are similar without any need for influence, because they are both Indo-European, but Roman religion owes a LOT to the very strange and non-IE influence of the Etruscans. Early Roman religion is much more difficult to talk about, partly because the literary sources are lousy, partly because the archaeology of early Italic cult is complicated, and partly because for Roman religion in particular, things quickly stray into obtuse anthropological jargon. Certainly by the 5th century BCE, Roman "religion" is already the fully-formed set of practices familiar to the Republic and Empire.
Some light reading:
Bremmer, Greek Religion (1994).
Dumezil, Archaic Roman Religion (1996).
Nilsson, A History of Greek Religion (1949 - oldie but goodie).