There are a lot of good answers already, but here's another point: a great way to reveal details is conflict. Apart from giving characters a good excuse to remind each other of something they all know, it automatically drives the plot and fleshes out the characters. Let's look at a few examples of this in action, all taken from Disney's Hercules. It's a film that had to explain a lot of Greek myth (albeit their modification of it) to the audience in a short time.
* The film begins with the <PERSON> telling a narrator they'll take over from him because he's too austere. This allows them to introduce their more fun and song-based style of narration.
* There is also conflict between individual <PERSON> - nothing too harsh or verbose, but enough you get a feel for their individual personalities, and how it'll factor into both spoken and sung narration.
* In song they blitz through the Titanomachy to the Olympian era, both revealing a conflict between two groups of gods and setting us up for the film's entire plot.
* As we get accustomed to the unique biological rules on Mount Olympus, <PERSON> comes in and you can feel the tension in the air. He doesn't glow like his fellow immortal gods, which makes his villainy more obvious. But his dialogue with the others quickly slips in his resentment over being forced to reside in the underworld, and how he wishes <PERSON> could somehow die.
* Next <PERSON> goes to the underworld and we quickly see him rub up against dead souls, his staff and the Fates. Each instance tells you something else about the world. They don't need to pause everything and say, "these are the Fates, who know all and direct all". Instead we get the hilarious "I KNOW you know, I got it, I got the concept" scene, while <PERSON> frustatedly tries to have a normal conversation about - you guessed it - his conflict with <PERSON>.
* I'll skip a minute or so of plot to where <PERSON> orders <PERSON> and <PERSON> to abduct the infant <PERSON>, turn him mortal and kill him. <PERSON> and <PERSON> don't have identical views, and the conflict between them tells you a little about their relationship with each other and <PERSON>. And the when/if <PERSON> finds out exchange sets us up for the dénouement, with the matter of when/if <PERSON> gets out of the Styx.
* A minute or so later, we skip to an adolescent <PERSON>, who is clearly very distrusted in his community, and it doesn't take long to find out why. 952
"<PERSON>" accidentally breaks everything, eventually leading the townspeople to turn on him and his human adoptive father <PERSON>. At this point we've already heard from the <PERSON> about <PERSON>'s great strength, but now we see what that means in his life. It also drives <PERSON> and his wife <PERSON> to reveal his connection to the Olympians, and that sets the next part of the plot in motion.
* <PERSON> has to convince <PERSON> to train him. <PERSON>'s having none of it. He's been disappointed before by <PERSON>, <PERSON> and "a lot of -euses". It takes <PERSON> striking him with lightning before he's convinced to help.
* <PERSON>'s introduction shows her in conflict with several characters in quick succession: the river guardian, <PERSON>, Pain, Panic and <PERSON>. Each exchange reveals different information about him, and sets up dramatic irony (i.e. we know she's working with the villains, but <PERSON> doesn't). It also involves conflict between <PERSON> and his other minions, when he discovers <PERSON>'s alive and they've kept it from him until almost the deadline for killing him.
* <PERSON> wants hero work, but the people of Thebes are as disinterested as <PERSON> had been. You quickly learn they've endured many trials, and amateur heroes have mishandled them so often they won't give a newbie a chance. It takes a prolonged and difficult defeat of a Hydra for them to change their minds. The conflict here is not between <PERSON> and the Hydra; it's between him and the townspeople.
Other conflicts occur throughout the film, but I won't mention them because by this point the worldbuilding is complete (save for occasional touches, e.g. 190