Transform.position in local space.

function Update () {

    if (Input.GetKeyDown ("up")) {
        transform.position.x += 1;

The object translates in the world axis instead of the local axis. I want it to move in its local axis instead of the world axis. Also, localPosition doesn't work since it isn't a child object. Help, please.

You can use transform.up, right and forward:

transform.position += transform.right;


transform.Translate(1, 0, 0);