How seriously did the Greeks/ Romans take their Gods?

I know the worship of Dionysus was fairly widespread and extremely fervent but what about the other gods? Also did they really believe their Myths?

It depended on the individual. Some of them believed every myth as the literal truth; others, especially later on, and especially the educated ones, were either atheists or Deists. For example, Aristotle believed in a God that was the Prime Mover after which all other things moved, but he didn’t think that God was like a person or anything. On the whole, though, they were pretty tolerant of other peoples’ beliefs. The problem with Christianity was that the Christians absolutely refused to participate in worship or sacrifices, which made the Romans fear that the Gods would punish the whole community for it. Religion at the time was a far less personal affair, and gods weren’t necessarily considered good, just beings that you didn’t want to tick off.

Vox Imperatoris

ETA: The Romans also had serious respect for tradition. That’s why Judaism, for the most part, escaped persecution, while the upstart Cult of Jesus was suppressed as much as possible. Both of these gods, too, were pretty innovative in that they actually loved you personally and unconditionally; the Romans did not think this at all.

Yes, I assume the gods did believe their myths; otherwise they couldn’t exist!

More seriously, this question is so enormously complicated as to be unanswerable, in part because of variation over the centuries. But for a few highlights:

Myths are meant to be believed, but not taken literally. Think about Judeo-Christian myths, the sacred narratives in the Bible. Some people take the stories literally. Others take them literally, with caveats: things don’t happen that way anymore, God doesn’t work casual miracles like in the Old Testament, or something. Many others consider them important, deeply meaningful stories but don’t worry about their truth value. Still others accept that the stories are not literally true, but consider them important. Finally, there are people who consider them ignorant, silly hogwash that get in the way of true religiosity. There is no reason to think that the Ancient Greeks were much different.

There are some important differences, too. The primacy of oral tradition meant that most religious knowledge was local and changeable. Religion a whole complex of narrative (myth), belief, ritual, festival, etc. These would have broad commonalities but endless variation.

There was no central authority, either textual or priestly, that held sway everywhere.

In a polytheistic system, you just can’t have a personal relationship with every deity (expensive and exhausting), so your investment in every aspect of the system is going to be at different levels. You may have a deep relationship with Poseidon, and thus a deeper investment in his myths, without really needing an opinion on whether Daphne was really turned into a laurel tree. Usually, individuals would have a ritual relationship with the gods important to them, and leave most of the day-to-day management to the priests of that god or goddess.