Change:
$goal = get_category_by_slug('goal');
foreach($categories as $category)
{
$output .= '<a href="'.get_category_link($category->term_id ).'" title="' . esc_attr( sprintf( __( "View all posts in %s" ), $category->name ) ) . '">'.$category->cat_name.'</a>'.$seperator;
}
Into:
foreach( $categories as $category ) {
if ( $category->parent == $goal->term_id)
$output .= '<a href="'.get_category_link($category->term_id ).'" title="' . esc_attr( sprintf( __( "View all posts in %s" ), $category->name ) ) . '">'.$category->cat_name.'</a>'.$seperator;
}
Where you replace '123'
with the ID of the Goal category.