Why do teaching and organizing chores fall to the pigs, and what is the first indication that they will use this to set themselves apart?
Because the pigs are smarter than the other animals (they can read), the other farm animals readily accept the fact that they take on a leadership position. Also, they have already a place of respect because of Old Major, the boar that incited the farm to rebellion against Farmer Jones. Indeed, many of the pigs were the offspring of Old Major. In his speech in the barn he declares:
I feel it my duty to pass on to you such political wisdom as I have acquired. I have had a long life I have had much time for thought as I lay alone in my stall, and I think I may say that I understand the nature of life on this earth as well as any animal now living.
The first sign of an oligarchy taking shape is when the pigs move into Farmer Jones' house and start adjusting the rules to fit their own comfort; for example, "No animal shall sleep in a bed" becomes "No animal shall sleep in a bed with sheets." Then Squealer, appointed to represent Napoleon, explains to the rest of the animals that the extra portion of milk and apples set aside for the pigs is to enhance their thinking abilility so that their intelligence will be opitmal for running the farm (and so that Jones will not come back).This kind of rationalization shows how the animals can be duped because of their basic ignorance.
On the side, Napoleon has also taken away Bluebell's puppies, that have been put in the barn to receive a "special training." Later in the story the reader learns that they have been conditioned to be killer dogs and function as Napoleon's secret police (hitherto, no longer a 'secret' to anybody!) By the time some of the animals wake up to what is going on, it is already too late to intervene.