Yes, there's actually quite a bit of evidence that the Greeks and Romans took their religion very seriously.

I'm more familiar with the Romans than the Greeks -- so sorry to essentially ignore half of your question but:

Until the Empire embraced Christianity in the 4th Century, the main Roman Religion was based of the Greek deities. But religion in Rome was more focused on the correct practice of prayer and rituals and not so much on the actual dogma of the religion (i.e. you didn't necessarily need to know who all the children of Jupiter were, or what there significance was, so long as you said your prayers correctly).

The Romans did have a habit of adopting Gods from other cultures that they conquered, which might make it seem like they didn't take religion all that seriously, but that was a tactical decision -- adopt their gods and they'll behave.

There was also a ton (40, to be exact) religious holidays a year and Priesthood was an important title for the Romans.

And, for a little bit of Greek knowledge, it's believed that Sappho, the lyrical poet from Lesbos was in a cult of Aphrodite and she wrote songs to her, like this:

Iridescent-throned Aphrodite, deathless Child of Zeus, wile-weaver, I now implore you, Don't--I beg you, Lady--with pains and torments Crush down my spirit, But before if ever you've heard my pleadings Then return, as once when you left your father's Golden house; you yoked to your shining car your Wing-whirring sparrows; Skimming down the paths of the sky's bright ether On they brought you over the earth's black bosom, Swiftly--then you stood with a sudden brilliance, Goddess, before me; Deathless face alight with your smile, you asked me What I suffered, who was my cause of anguish, What would ease the pain of my frantic mind, and Why had I called you To my side: "And whom should Persuasion summon Here, to soothe the sting of your passion this time? Who is now abusing you, Sappho? Who is Treating you cruelly? Now she runs away, but she'll soon pursue you; Gifts she now rejects--soon enough she'll give them; Now she doesn't love you, but soon her heart will Burn, though unwilling." Come to me once more, and abate my torment; Take the bitter care from my mind, and give me All I long for; Lady, in all my battles Fight as my comrade.

So basically, some people took one particular god very seriously, while others just focused on knowing how to honor them -- people weren't too concerned with the soap-opera taking place on Mount Olympus

(Also -- slightly off topic but there were other cults and religions that floated around Rome, the most interesting I think is Mithraism, worshipped by Roman Soldiers. I had a book on them, but I'm in Vietnam right now and don't have my books handy, so here: https://en.wikipedia.org/wiki/Mithraism)