Why do the teaching and organizing chores fall to the pigs, and what is the first indication that they will use this to set themselves apart?
The pigs were given the task of organizing the new society because they were known to be "the cleverest of the animals." This makes sense in terms of the socialist allegory Orwell is writing. The intellectuals are the vanguard of the revolution, preparing the way for the less enlightened proletariat (workers), who are in this case the other animals. However, one might wonder if they also take the task on themselves as a commentary: they are greedy, and human-like in their desire to consume all. This makes them a good metaphor for capitalism.
The first indication they will use this to set themselves apart is the comment that Squealer could turn black to white with his words.