Echoing Categories only if they’re a child of a given category

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.