Code for movement controller of the player in Unity
First of all, I didn't like the long method names you used in the script. 
like:

void Start ()
{
    initializeNeededForMovementVariables();
}


I mean, why did you need to call it like that? In my opinion this makes the code very messy once you have a lot of code in a script.
I would had call it something like: "InitVariables" or simply "Init"

Again, 

float moveHorizontallyValue = Input.GetAxis("Horizontal");


The same point applies here too. You could have just written:

float h = Input.GetAxis("Horizontal");


Another thing: Why are you doing the following block?

private void playAnimationOfHorizontalMovement()
{
    if (currentSpeed == walkingSpeed)
    { playerAnimationController.beingOnlyWalking(); }
    else if(currentSpeed == runningSpeed)
    { playerAnimationController.beingOnlyRunning(); }
}


You have all this code to play animations. I suppose you are using other scripts too, and again, this is my opinion, it makes everything not totally understandable.

Then, Why aren't using an animator?

The code would be A LOT less and also A LOT more understandable, plus WAY EASIER to code!

Now, don't worry if your code isn't perfect now, you'll get better at it. It requires time. I really hope this helps you, Let me know!

Cheers