why does what get pushed away when centripetal acceleration is towards the center If centripetal acceleration is towards the center, then why - when you spin a bucket of water (a classic demonstration) - does the water not get pushed out but rather stays in the bucket without spilling?
 A: Nothing gets "pushed away". Instead, if it was left to itself everything would fly off in a straight line (Newton's law, right?).
This is easy to see if you are, say, hovering over the playground in a helicopter watching children fly off the merry-go-'round. 
I did a sketch. The dark haired kid is holding on; he goes in a circle. The blond it not; he moves at a constant velocity. 

Remember that acceleration in a change in velocity and that velocity means both speed and direction.
So, the key question is why does it look like they are getting "pushed away"? The answer is that when you see that happening you are getting pushed toward the center (that's the centripetal force). You are accelerating, while they maintain a constant speed and direction, but they certainly end up a long way away from you.
A: For the same reason you are "pushed" backward when your car accelerates in the forward direction.
Let us analyze the situation from an external inertial observer's point of view. The centripetal force acts on the bucket towards the center, but the water due to its inertia "tries" to maintain its state of motion. It appears as if the water gets pushed away from the center.
More generally whenever a reference frame gets accelerated (becomes non-Newtonian) any object within it "feels" a "force" in the opposite direction of the acceleration. For an outside inertial observer this is simply the result of the inertia of the objects within the accelerated frame.
However, more generally, following "the principle of equivalence" all observers can be given equal status and it is always possible for an observer in the accelerated frame (who/ which should be very very small) to consider himself/ herself stationary and claim that there is a gravitational field which is responsible for the apparent non Newtonian behaviors.
A: It is possible to accelerate towards a point, without moving any closer to the point, you just need to circle around the point.
When you are inside a bucket, moving near the bucket wall, constantly colliding with the wall, then you are constantly accelerating away from the wall, but staying near the wall.
There is no fundamental difference between moving around in a perfect circle in a bucket, and bouncing around randomly in a bucket. In both cases you collide with the wall, the wall pushes you inwards, you push the wall outwards.  
A: The reason the water doesn't fall out of the bucket (when moving in a vertical circle) is because you have thrown it in the air at the bottom of the swing, the trick is then to 'pull' it back down quickly and then 'catch' it again in the bucket.  The apparent pull of the water upwards towards the top of the swing is simply the force associated with the inertia of the water,  it is travelling upwards at some speed and you are accelerating it downwards at some acceleration...
A: An object can accelerate towards the center without actually ever moving towards the center because of net force acting on the object. The net force is an unbalanced force acting on an object so when the ball's acceleration pushes towards the center the net force pushes the ball to the outside.  
