带有验证码的php表单发送给godaddy电子邮件而不是gmail

I made a php form from info on the web, and it will only work if i send it to a godaddy email that I have made. I need to transfer the email from godaddy to gmail but no luck. It also includes a captcha that is put in correctly just not working :/. Would be very much appreciated if someone could check the code:
PHP:

<?php
$field_name = $_POST['name'];
$field_email = $_POST['email'];
$field_phone = $_POST['phone'];
$field_subject = $_POST['subject'];
$field_message = $_POST['message'];

$mail_to = 'blah@abc.org';
$subject = $field_subject;

$body_message = 'From: '.$field_name."
";
$body_message .= 'E-mail: '.$field_email."
";
$body_message .= 'Phone: '.$field_phone."
";
$body_message .= 'Message: '.$field_message;

$headers = 'From: '.$field_email."
";
$headers .= 'Reply-To: '.$field_email."
";

$mail_status = mail($mail_to, $subject, $body_message, $headers);

    if(isset($_POST['g-recaptcha-response'])&& $_POST['g-recaptcha-response']){
        var_dump($_POST);
        $secret = "6LfjxQgUAAAAAGtDuv9q3nnhCEOrkrP2ywsQoL9k";
        $ip = $_SERVER['REMOTE_ADDR'];
        $captcha = $_POST['g-recaptcha-response'];
        $rsp  = file_get_contents("https://www.google.com/recaptcha/api/siteverify?secret=$secret&response=$captcha&remoteip$ip");
        var_dump($rsp);
        $arr = json_decode($rsp,TRUE);
        if($arr['success']){ ?>
                <script language="javascript" type="text/javascript">
                      window.location = 'index.html';
                   </script>
            <?php
        }else{ ?>
                <script language="javascript" type="text/javascript">
        window.location = 'index.html';
       </script>
        <?php
        }

    }

HTML:

<form id="contact-form" action="contact-form.php" method="POST">
                            <div class="inline-inputs">

                                <div class="col-lg-12 col-md-12 col-sm-12">
                                    <label>* Name</label>
                                    <input type="text" name="name">     
                                </div>

                                <div class="col-lg-7 col-md-7 col-sm-12">
                                    <label>* Email address</label>
                                    <input type="text" name="email">                                
                                </div>

                                <div class="col-lg-5 col-md-5 col-sm-12">
                                    <label>Phone</label>
                                    <input type="text" name="phone">    
                                </div>

                                <div class="col-lg-12 col-md-12 col-sm-12">
                                    <label>* Subject</label>
                                    <input type="text" name="subject"> 
                                </div>

                                <div class="col-lg-12 col-md-12 col-sm-12">
                                    <label>* Your questions/comments</label>
                                    <textarea rows="10" name="message"></textarea>
                                </div>

                            </div>
                            <div class="g-recaptcha" data-sitekey="6LfjxQgUAAAAAOHsDVpGTeYNUoj2snz_dI_8SeH5"></div>
                            <input type="submit" value="submit">

                        </form>

(I'm also very new so excuse the bad question and the lack of knowledge)