What's Euripides's view on religion?

Euripides's view of religion is considerably more nuanced than that of many of his fellow Athenians. For one thing, he displays a fair degree of skepticism throughout his works, not just in relation to the gods, but towards certain key elements of ancient Greek religion. The notion of blind faith, though common at the time, is not one that appears to be shared by Euripides. Nor would it seem that he has much time for soothsayers and prophets, whose words were treated by the ancient Greeks with such astonishing reverence.
Yet it is the gods of Olympus who are the main target of Euripides's withering skepticism. With good reason, Samuel Taylor Coleridge described Euripides as never being more happy than when he was administering a great slap to all the gods. His portrayal of them is less than flattering, to say the least. They are greedy, lustful, treacherous, and vain, treating men like mere playthings as part of a gigantic cosmic game whose rules have been rigged to favor the immortals. In Ion, the lead character chastises the gods for their shameless double-dealing and hypocrisy:

How is it then just that you, who gave the laws to mortals, should yourselves transgress those laws?

Euripides does not call into question the existence of the gods as such. He does, however, question whether they deserve to be worshiped, given their appalling behavior. Implicit in Ion's lament is the assumption that gods ought to lead by example if they are to earn the respect, as opposed to fear, of those who worship them.
In Herakles Euripides allows Amphitryon to voice an even more bitter condemnation of the gods and their destructive impact on mortal man:

Even I, though but a mortal, surpass thee in virtue. I have not betrayed the sons of Hercules. As for thee, thou knewest how to steal in secret to the bed of another's wife, but to save thy friends thou knowest not. A feeble god, or a treacherous, art thou.

Even more boldly, Hercules implies that a god who doesn't behave in a morally exemplary manner isn't really a god at all:

Never will I believe, that the gods pursue unlawful loves, or that they throw their kindred into bonds, or that one is master of the other. God, if he be a true god, has need of nothing. These are but the pitiful fables of poets.

Yet that is precisely how the gods do behave in Greek mythology, time and time again. The gods are being measured against human standards and found wanting, not just in relation to their morals, but even their very status as immortals.
Having said that, Euripides, like the philosopher Plato, understands the overriding value of public religion for the good order and long-term stability of Athens. But it's the nature of that religion, its rites, beliefs, and practices, which is of cardinal importance for Euripides. And it is here that he shows a consistent hostility to the various elements of superstition, folklore, and myth that have accrued to the city-state's religious life over the course of time.
Euripides harbors a particular animus towards oracles, seers and soothsayers of various kinds. His motives would appear to be political. Athens' defeat in the Peloponnesian War led to a huge backlash against the use of divination to determine the timing and conduct of military expeditions. In common with many of his fellow Athenians, Euripides came to see the whole rich panoply of superstitions as tending to the destruction of the polis. Once again, Euripides appears to suggest that humans must look to themselves to determine what ought to be done, instead of wasting their time consulting oracles or worshiping unworthy, amoral deities. A classic statement of this proto-humanistic concept is provided in Helen:

False and worthless are the utterances of soothsayers, nor is wisdom to be found in flames of fire, or in the voices of the feathered tribe. 'Tis folly to hope that birds can bring benefit to mortal men.... Let us rather, at our sacrifices, beseech the gods to send us blessings, and let us pay no heed to oracles.... Wisdom and prudence are the wisest soothsayers.

As we said at the outset, however, Euripides's approach to religion is nuanced. His skepticism, though significant, is far from destructive. Indeed, in plays such as The Bacchae and Hippolytus Euripides appears keen to highlight the folly of intellectual pride and of turning one's back on established religious norms. In The Bacchae, for instance, King Penteus's skepticism regarding the propriety of Dionysiac ritual leads directly to his own gruesome death.
In his famous comedy The Frogs, Aristophanes has a lot of fun at the expense of Euripides, as he does with all the great Athenian tragedians. He satirizes him as someone who abuses his undoubted gifts as a dramatist to try and convince the people that there are no gods. Yet this is no more than a comic exaggeration. There's no evidence to suggest that Euripides denied the gods' existence. If he seriously had, then his plays would simply not have been staged in the first place.
However, there can be no doubt that Euripides encouraged his fellow Athenians to examine more closely the nature of their beliefs, while at the same time maintaining an appropriate degree of respect for the established religious rites and practices, even if only from a utilitarian perspective. Fables, myths, and stories may not literally be true, suggests Euripides, but they do still serve an important purpose, nonetheless. If they conduce to the well-being and stability of the city-state, all well and good. If they do not, however, then they must be exposed to contempt and ridicule in the hope that they will be purged altogether. In other words, for Athenian public religion to be preserved and maintained it is necessary for it to develop a more questioning, self-critical edge. And it is that which Euripides, more than any of the great tragedians of antiquity, provides in abundance.