problem with character motion and movement

Usually using CharacterControllers is the best thing to do. You can start with the CharacterController.Move example of movement script - it's easy to understand and modify. Just to have an idea, the script below is the example script with run and mouse control to turn left/right:

var walkSpeed : float = 6.0; var runSpeed : float = 9.0; var turnSpeed : float = 90; // speed to turn left/right

var jumpSpeed : float = 8.0; var gravity : float = 10.0;

private var moveDirection : Vector3 = Vector3.zero; private var angles : Vector3 = Vector3.zero;

function Update() {

 var controller : CharacterController = GetComponent(CharacterController);
 
 // turn left/right (around Y) with mouse X
 angles.y = (angles.y + Input.GetAxis("Mouse X")*turnSpeed*Time.deltaTime) % 360;
 transform.localEulerAngles = angles;

 if (controller.isGrounded) {
     moveDirection = Vector3(Input.GetAxis("Horizontal"), 0,
                             Input.GetAxis("Vertical"));
     moveDirection = transform.TransformDirection(moveDirection);

     // define speed
     var speed = walkSpeed; // assume walkSpeed, but if shift pressed...
     if (Input.GetKey("left shift")) speed = runSpeed; // change to runSpeed
 
     moveDirection *= speed;
     if (Input.GetButton ("Jump")) {
         moveDirection.y = jumpSpeed;
     }
 }
 // Apply gravity
 moveDirection.y -= gravity * Time.deltaTime;
 // Move the controller
 controller.Move(moveDirection * Time.deltaTime);

}