How can I get PHP form submissions to be sent to my gmail?

Question

I created a simple form in an HTML document, which links to a PHP document. The php document is supposed to forward the form to my email. It's a gmail address. I'm using godaddy cpanel. I do not receive any email. I need to fix the code and/or settings in godaddy or gmail so that I can receive this form data.

All the below HTML and PHP code I found online from someone that says it works. So therefore I would definitely appreciate the entire code that you know will work.

I am not an expert in html or php or their terminology, so please use non-technical words.

I already added the gmail MX records on godaddy's cpanel for this domain. I also changed in Godaddy the email settings from "local email exchanger" to "remote mail exchanger" (so that godaddy doesn't try to use only their internal software).

These are the two basic things that I've found online that supposedly should handle the problem.

THIS IS THE FORM IN THE HTML DOC:

<form name="contactform" method="post" action="send_form_email.php">
    <table width="450px">
        <tr>
            <td valign="top">
                <label for="first_name">First Name *</label>
            </td>
            <td valign="top">
                <input  type="text" name="first_name" maxlength="50" size="30">
            </td>
        </tr>
        <tr>
            <td valign="top"">
            <label for="last_name">Last Name *</label>
            </td>
            <td valign="top">
                <input  type="text" name="last_name" maxlength="50" size="30">
            </td>
        </tr>
        <tr>
            <td valign="top">
                <label for="email">Email Address *</label>
            </td>
            <td valign="top">
                <input  type="text" name="email" maxlength="80" size="30">
            </td>
        </tr>
        <tr>
            <td valign="top">
                <label for="telephone">Telephone Number</label>
            </td>
            <td valign="top">
                <input  type="text" name="telephone" maxlength="30" size="30">
            </td>
        </tr>
        <tr>
            <td valign="top">
                <label for="comments">Comments *</label>
            </td>
            <td valign="top">
                <textarea  name="comments" maxlength="1000" cols="25" rows="6"></textarea>
            </td>
        </tr>
        <tr>
            <td colspan="2" style="text-align:center">
                <input type="submit" value="Submit">
            </td>
        </tr>
    </table>
</form>

---

THIS IS THE PHP DOC "send_form_email.php"

<?php
if(isset($_POST['email'])) {

    // EDIT THE 2 LINES BELOW AS REQUIRED
    $email_to = "myname@gmail.com";
    $email_subject = "submission_form";

    function died($error) {
        // your error code can go here
        echo "We are very sorry, but there were error(s) found with the form you submitted. ";
        echo "These errors appear below.<br /><br />";
        echo $error."<br /><br />";
        echo "Please go back and fix these errors.<br /><br />";
        die();
    }


    // validation expected data exists
    if(!isset($_POST['first_name']) ||
        !isset($_POST['last_name']) ||
        !isset($_POST['email']) ||
        !isset($_POST['telephone']) ||
        !isset($_POST['comments'])) {
        died('We are sorry, but there appears to be a problem with the form you submitted.');
    }



    $first_name = $_POST['first_name']; // required
    $last_name = $_POST['last_name']; // required
    $email_from = $_POST['email']; // required
    $telephone = $_POST['telephone']; // not required
    $comments = $_POST['comments']; // required

    $error_message = "";
    $email_exp = '/^[A-Za-z0-9._%-]+@[A-Za-z0-9.-]+\.[A-Za-z]{2,4}$/';

    if(!preg_match($email_exp,$email_from)) {
        $error_message .= 'The Email Address you entered does not appear to be valid.<br />';
    }

    $string_exp = "/^[A-Za-z .'-]+$/";

    if(!preg_match($string_exp,$first_name)) {
        $error_message .= 'The First Name you entered does not appear to be valid.<br />';
    }

    if(!preg_match($string_exp,$last_name)) {
        $error_message .= 'The Last Name you entered does not appear to be valid.<br />';
    }

    if(strlen($comments) < 2) {
        $error_message .= 'The Comments you entered do not appear to be valid.<br />';
    }

    if(strlen($error_message) > 0) {
        died($error_message);
    }

    $email_message = "Form details below.\n\n";


    function clean_string($string) {
        $bad = array("content-type","bcc:","to:","cc:","href");
        return str_replace($bad,"",$string);
    }



    $email_message .= "First Name: ".clean_string($first_name)."\n";
    $email_message .= "Last Name: ".clean_string($last_name)."\n";
    $email_message .= "Email: ".clean_string($email_from)."\n";
    $email_message .= "Telephone: ".clean_string($telephone)."\n";
    $email_message .= "Comments: ".clean_string($comments)."\n";

    // create email headers
    $headers = 'From: '.$email_from."\r\n".
        'Reply-To: '.$email_from."\r\n" .
        'X-Mailer: PHP/' . phpversion();
    @mail($email_to, $email_subject, $email_message, $headers);
    ?>

    <!-- include your own success html here -->

    Thank you for contacting us. We will be in touch with you very soon.

    <?php

}
?>