Nearly all physics problems will use the unit m/s^2 for acceleration. explain why the seconds are squared. why isn't the unit given as m/s, as it is for speed

Because acceleration is not speed, and speed is not acceleration. 
I'm sure you would not ask "Why isn't temperature given in acres,
as it is for area ?"  Speed and acceleration are different things, so
it's only natural that they have different units.

The magnitude (size) part of acceleration is:

         (how much speed changes) per second.

Are you speeding up ?
Are you traveling (2 meters per second) faster every second ?
Then your acceleration is

           (2 meters per second) per second .

When you write that phrase as an algebraic expression, it's

           (2 m/sec) / sec

and when you simplify that fraction, you get        2 m/sec² .

Independent variable is something that is changed in a experiment.