Which is the world's oldest religion? What is the evidence of it?

We don't know what is the oldest religion. It is quite possible that such a religion is long extinct. Religious practices are found even in urban agglomerations found 10000 years ago such as Çatalhöyük. Sumerians also had literature such as Kesh temple hymn that are dated to 2600 BCE. Those religions are all long gone. 

Among the existing major religions the oldest is Hinduism. Vedas form the oldest body of religious works that are still in common use. Rigveda is dated to 1500 BCE or earlier and is used in the exact same form to date. It is likely that the religious ideas rose much earlier as complex literature usually follows basic religious structures.  In parallel, we have more archaelogical evidence from the Middle Eastern Mittani kingdom that used the same gods as the Vedas (http://www.jstor.org/stable/595878). Hebrew ideas also likely arose from that period, but it was not until the 5th century BCE work of Torah did it get a more religious form. 

There is still some controversy on the origins of Jainism. Historians generally believe Mahavira to be the founder of the modern Jainism. We don't have any Jain agamas older than 6th century BCE. However, those religious texts allude to 23 religious leaders called the Tirthankara who came before Mahavira. Assuming that those gurus came one per generation, the first of the Tirthankaras could have come as early as 1200 BCE. As we unearth the past of India, we might find both Hinduism and Jainism much older than what we currently assume.