how does redshift prove expansion is accelerating? My astronomy teachers never would answer this for me... Redshift obviously indicates an object (such as another galaxy) is moving away, but how do we know its acceleration from this? It's my understanding that the instantaneous redshift of an object is an indicator solely of its instantaneous velocity, and we have certainly not been observing redshift long enough to compare and determine a change in velocity

If you can, please avoid using complex math in your answer.
You're right, a redshift will give you a measure of a velocity, not an acceleration. I'm assuming when you say acceleration, you're referring to the accelerating expansion of the universe so you're not looking at the particular acceleration of an object (which you could get from examining the force of gravity imparted on that object). 

So in addition to a redshift, you need to find a way to independently measure the distance. To get out to an appreciable redshift (z~0.5), Type Ia supernova are used as "standard candles". They are considered standard because they're believed to be from white dwarfs that explode because they have reached their maximum possible mass, the Chandrasekhar limit, of 1.4 solar masses. Therefore, when they explode, they should all look about the same and any difference in brightness would be from how far they are away from us. So they measured the distance and the redshift of the galaxy the supernova originated from to determine the expansion of the universe appears to be accelerating over time.