Custom registration form

Hello, i’ve create this custom register form

<!-- Row for main content area -->
    <div id="content" class="left eight columns" role="main">
    <div class="post-box">
                    <?php get_template_part('/includes/content', 'page'); ?>
        <div class="wrapper">
<?php
$err = '';
$success = '';

global $wpdb, $PasswordHash, $current_user, $user_ID;

if(isset($_POST['task']) && $_POST['task'] == 'register' ) {

    $pwd1 = $wpdb->escape(trim($_POST['pwd1']));
    $pwd2 = $wpdb->escape(trim($_POST['pwd2']));
    $first_name = $wpdb->escape(trim($_POST['first_name']));
    $last_name = $wpdb->escape(trim($_POST['last_name']));
    $email = $wpdb->escape(trim($_POST['email']));
    $username = $wpdb->escape(trim($_POST['username']));

    if( $email == "" || $pwd1 == "" || $pwd2 == "" || $username == "" || $first_name == "" || $last_name == "") {
        $err = 'Devi compilare tutti i campi obbligatori';
    } else if(!filter_var($email, FILTER_VALIDATE_EMAIL)) {
        $err = 'Indirizzo email non valido.';
    } else if(email_exists($email) ) {
        $err = 'Indirizzo email gi&agrave registrato.';
    } else if($pwd1 <> $pwd2 ){
        $err = 'Le password non corrispondono.';
    } else {

        $user_id = wp_insert_user( array ('first_name' => apply_filters('pre_user_first_name', $first_name), 'last_name' => apply_filters('pre_user_last_name', $last_name), 'user_pass' => apply_filters('pre_user_user_pass', $pwd1), 'user_login' => apply_filters('pre_user_user_login', $username), 'user_email' => apply_filters('pre_user_user_email', $email), 'role' => 'subscriber' ) );
        if( is_wp_error($user_id) ) {
            $err = 'Errore in fase di registrazione';

        } else {
            do_action('user_register', $user_id);

        $success = 'You\'re successfully register';
             wp_set_current_user( $user_ID, $current_user );
             do_action('set_current_user');
             $redirect_to = site_url('www.google.it');

             wp_safe_redirect($redirect_to);
        }

    }

}
?>

    <!--display error/success message-->
<div id="message">
    <?php
        if(! empty($err) ) :
            echo '<p class="error">'.$err.'</p>';
        endif;
    ?>

    <?php
        if(! empty($success) ) :
            echo '<p class="error">'.$success.'</p>';
        endif;
    ?>
</div>

<form method="post">
    <h3>Registrati al Portale AlpeAdria</h3>
    <div class="customregistrazione">
    <label class="customregistrazione">Cognome*</label>
    <input type="text" value="" name="last_name" id="last_name" style="width: 50%;" />
    <label >Nome*</label>
    <input type="text" value="" name="first_name" id="first_name" style="width: 50%;" />
    <label>Email*</label>
    <input type="text" value="" name="email" id="email" style="width: 50%;" />
    <label>Username*</label>
    <input type="text" value="" name="username" id="username" style="width: 50%;" />
    <label>Password*</label>
    <input type="password" value="" name="pwd1" id="pwd1" style="width: 50%;" />
    <label>Reinserisci Password*</label>
    <input type="password" value="" name="pwd2" id="pwd2" style="width: 50%;" />
    </div>
    <div class="alignleft"><p><?php if($sucess != "") { echo $sucess; } ?> <?php if($err != "") { echo $err; } ?></p></div>
    <div>Compilando il seguente form di registrazione il cliente accetta quanto previsto nel disclaimer</div>
    <button type="submit" name="btnregister" class="button" >Registrati</button>
    <input type="hidden" name="task" value="register" />
</form>
</div>

    </div><!-- End Content row -->

    <?php get_sidebar( 'right' ); ?>
<?php get_footer();?>

And i want that the user redirect when the registration is ok. Now when visitor make a registration have the succesful message and continue see the form. I like automathic redirect registration to define page http://www.mysite.it/page. Whit my code the form function but not the redirect.

Also i like send an email to subscriber user with details

I try to add this but not function email ($email, "Registrazione OK", "Complimenti registrazione effettuata con successo", "From: email@redacted.com");

Where wrong? How can i resolve?

Thank’s