Vector3.Slerp leads to wrong axis rotation

I need some help with my character movement function. I need my character (for a 3D platformer) to smooth its turns, preferably to the point that the character runs in a perfect circle where a character would normally walk in a square. After scanning the Answers and Forums, I discovered how to use the Slerp() method to smooth rotations, but my result is always different from the results given by other users.

Whenever I start my test, my character is face-down (rotated on the X axis by 90 degrees). Also, when I press a key and release it, the character keeps moving in that same direction (probably caused by the apparent misuse of the Slerp() method).

Here's my function. I can't find anything here that would cause that, so an extra set of eyes would be very helpful. Why won't my solution work?

 //  This handles movement and rotation.
 void BasicMovement()
 {
     CharacterController controller = GetComponent<CharacterController>();
     Vector3 targetDirection;

     //  If the player is touching the ground, move normally
     if (controller.isGrounded)
     {
         //  Take input from the keyboard.
         targetDirection = new Vector3(Input.GetAxis("Horizontal"), 0, Input.GetAxis("Vertical"));

         //  Attempt to smooth from original direction to new direction
         moveDirection = Vector3.Slerp(moveDirection, targetDirection, 1.0f * Time.deltaTime);

         //  Multiply by the move speed and add vertical velocity
         moveDirection = (moveDirection.normalized + new Vector3(0, verticalSpeed, 0)) * moveSpeed;

         //  If the player is moving then handle rotation
         if (moveDirection != Vector3.zero)
         {
             //  use SetLookRotation to make characters face where they're going. 
             Quaternion rotation = transform.rotation;
             rotation.SetLookRotation(moveDirection);
             transform.rotation = rotation;
         }
     }

     //  Only activate gravity if the character is in the air.
     if (!controller.isGrounded)
         moveDirection.y -= gravity * Time.deltaTime;

     //  Finally, move the character, according to its frame rate
     controller.Move(moveDirection * Time.deltaTime);
 }