There are examples of religious or at least ritual practice that go back many thousands if not tens of thousands of years. Sites like Gobekli Tepe (10th-8th mil BCE) are examples of early monumental religious organization but far before that we have potentially intentional burials dating to 90 thousand BP in the Qafzeh and Es Skhul Caves in Israel. However to what extent burials can imply religious practice is endlessly debatable and without writing there just isn't any way to determine what these people believed.

The first religious texts that give us an insight into the beliefs of people would be the various hymns of Sumer an example of which would be the Kesh Temple Hymn (2600 BCE) which along with the Instructions of Shuruppak is one of the oldest pieces of writing we have. It extols Enlil god of the wind and his daughter Nidaba goddess of writing and the harvest as well as Kesh and the great temple there. Another example would be the The Exaltation of Inana composed by Enheduanna daughter of Sargon the Great of Akkad in 2300 BCE.

We can trace Sumerian gods further back into the the Uruk (4000-3100 BCE) period. Inanna, goddess of fertility, war and patron of Uruk, for example appears on the Warka Vase (3200 BCE) and her symbol appears on the proto-literate seals of the Jemdet Nasr period (3100-2900).

Source The Electronic Text Corpus of Sumerian Literature

It's unknowable. Why? Even if we limit ourselves to "known religions", what we know diminishes we go further into the past. The oldest religion we know of existed somewhere 5,000 and 1.5 million years ago, depending on how we define things.

There's no universal definition of religion. We general define religion as a set of beliefs in something supernatural or a variation on that. Some emphasize ritual. Some emphasize transcendence. Some avoid all four, as Geertz did in his famous definition where he argued that religion was a collect of symbols:

(1) a system of symbols (2) which acts to establish powerful, pervasive and long-lasting moods and motivations in men (3) by formulating conceptions of a general order of existence and (4) clothing these conceptions with such an aura of factuality that (5) the moods and motivations seem uniquely realistic

But regardless of what definition of religion we choose, how do we expect those things to show up in the archeological record? For a full idea of a religion, we need writing to really describe beliefs, to get explanations of symbols. The first writing comes from Sumeria a bit over 5,000 years ago, thus perhaps it makes sense to grant Sumerians the first religion. /u/missingpuzzle discusses this. And if we limit ourselves more strictly, and ask what is the oldest religion that is still practiced today, we'd probably say either Judaism, Hinduism, or Zoroastrianism, but even this is complex (see here, as it's hard to even say that Judaism, Hinduism, or Zoroastrianism of two millennia ago are really undeniably the same as the Judaism, Hinduism, and Zoroastrianism of today).

But we have religious symbols older than writing that we generally would recognize as religion. These days, many people recognize Göbekli Tepe in Turkey (as /u/missingpuzzle also mentions) as the first major temple complex. The oldest stratum of Göbekli Tepe is 12,000 year old--7,000 years older than writing. There's a new and increasingly popular argument that, rather than the development of agriculture leading to increasing religious complexity, increasing religious complexity led to more concentrated masses of people which lead to agriculture. To read more about this, perhaps you'd be interested in Elif Batuman's article in the New Yorker: "The Sanctuary" or Charles C. Mann's piece in National Geographic magazine: "Gobekli Tepe: The Birth of Religion?" (see here for a summary of some of the challenges to the evidence for this the religion-->agriculture theory laid out in these articles, and an argument that these structure might not have even had a ritual purpose, and that either these structures themselves were residential, or that people lived elsewhere on site in a yet-to-be excavated area).

But before people made monumental religious structures (like Göbekli Tepe might be), did they have what we might recognize as religion? Probably. Göbekli Tepe is 12,000 years old. We have objects that were likely ritual objects much older than that. The famous Venus of Willendorf is likely older than 25,000 years old. Was this a ritual object, or was this just a decorative object or a toy? I'd be willing to say likely a ritual object, likely linked to fertility, but all we have is the object. Without writing, we can only guess its meaning. If I showed someone a Christian cross 25,000 years in the future, could they tell it was a religious object and not merely decorative object? Would an icon of Jesus be identifiably different from a picture of Elvis? Probably not. This is essentially half the problem we have with trying to connect paleolithic art and religion: did people do these for religious reasons, or did they just think they look nice? It's really, really difficult for us today to make that distinction, and most of these older objects are not as as clearly ritual objects as the Venus Willendorf is. It's unclear if the people living then would have made that distinction, between "religious" and "decorative". The other problem is we can't always distinguish what's intentional and what's just a coincidence of erosion--cf. the 300,000 year old "Venus of Tan-Tan". How do we know an object is a religious object if we have no context for it?

For me, where I would be willing to say "we have fairly good reason to suspect that something we'd broadly recognize as religion starts at least by this point" is intentional burial. This, in general, shows that there's likely some idea of an afterlife, especially if there are grave goods or decorations of the body and bones with things like flowers or red ochre (which we perhaps start finding tens of thousands if not hundreds of thousands years after we have evidence of intentional burial, depending on how one counts). I believe we start getting grave goods/decorations somewhere between 60-130,000 years ago, again depending on what counts. We're unclear on when we first have evidence of intentional burial. Somewhere between 90,000-300,000 years ago, was the thinking until recently--see /u/400-Rabbits's post from two year ago. But just this year archeologists found a site that, seemingly, is pretty clear evidence of intentional hominid (not even human) disposal of remains 1,500,000 years ago. But there are no goods here, only bones. But these hominids clearly put placed the remains of their dead there for a reason. Does this count as ritual behavior, as religion? I can't say.

So how old is religion? It depends almost entirely on what constitutes evidence of religion.

Here are two similar questions with extensive answers:

What is the earliest religion that we know about and what did they believe?

What is the world's earliest known form of religion and what did the group believe?

These are all excellent responses, but I am surprised to see no mention of "The Old Way", a narrative of Elizabeth Marshall Thomas' visit with the African bushmen in the 1950s and her mother's notes on the same. It is available on Amazon. In it she discusses the spoken religious traditions of the hunter gatherers that have no known origin, and never took written form. Perhaps this falls into the realm of mythology, but to the Bushmen it was their religion.

Edit The Old Way

Do you mean the oldest religion still being actively practiced, or the earliest religion we know of?

A more specific question; what's the oldest universal religion (a religion that actively seeks followers, rather than relying on ethnic ties)

I have various answers. First is Hinduism, with evidence going back to about 5,500 BC. Next is animism, which we can suspect goes back to the dawn of time by inference (tribal peoples are often animist, and historical religious scholars saw it as the most primitive of religions). Then there is the 70,000yr old snake cult of Ngamiland, Botswana.

Finally, evidence of burial goes back perhaps 300,000yrs, although one can debate regarding this being a religious ritual, a practical measure, or perhaps both.

Edit:

A controversial opinion regarding the religious symbolism of Homo rudolfensis/habilis some two million years ago.