newtonian mechanics - Can you completely explain acceleration to me?

I understand what acceleration is, and I know the formula, and I understand it's a vector. I just don't understand how the equation works exactly. I'm kind of picky, I know, but bear with me.

Velocity is the amount of distance traveled during the amount of time, $\frac{s}{t}$. That makes sense. But how on earth is acceleration related to the distance divided by time squared? Where does the squared come in?

I mean, yes, I can prove it mathematically. $\frac{\text{distance/time}}{\text {time}}$ is $\frac{\text{distance}}{\text{time}^2}$. But why? How is it possible to square time? Can I even assume to understand the individual components separately? Or do I have to assume I'm dividing a vector by time and just view it that way? Is it the amount of velocity absorbed during a different amount of time, or something?

I can't just plug in numbers and say I understand physics, even when I understand the end result. It's like A->B->C. I understand A and C, but where did the B come in? How was this proven? Maybe there's a proof online or something? All I could find was the proof for centripetal acceleration...

Basically what I'm asking is how each of the variables relate to one another separately, and how that all works out.

I really need to understand physics, or at least to the point where I can manipulate the equations in my mind to spatially transpose graphs of the accurate calculations on reality. I'm never going to make it over the hurdle in gym class if I can't even understand how fast I'm accelerating.

Answer

It's simpler than you (probably) think.

In your example of defining speed: this is a change of position $s$ in a time $t$. The units of distance are metres and the units of time are seconds, so the units of velocity are metres per second. So far so good.

Now consider acceleration: this is a change of velocity $v$ in a time $t$, so the units of acceleration are $v/t$. The units of velocity are metres/sec and the units of time are seconds, so the units of acceleration are (metres/sec)/sec or metres/sec$^2$.