Trying to create a form to send email with HTML and PHP

Question

I am building survey website for practice. I am trying to build a form and submit to gmail or yahoo.

The following is my HTML codes.

<form id="wrapped" method="POST" autocomplete="off">
    <input id="website" name="website" type="text" value="">
        <div id="middle-wizard">
            <div class="submit step">
                <h3 class="main_question">Please tell us how can we improve.</h3>
                <div class="form-group">
                    <label for="firstname">First Name</label>
                    <input type="text" name="firstname" id="firstname" class="form-control required">
                </div>
                <div class="form-group">
                    <label for="lastname">Last Name</label>
                    <input type="text" name="lastname" id="lastname" class="form-control required">
                </div>
                <div class="form-group">
                    <label for="email">Your Email</label>
                    <input type="email" name="email" id="email" class="form-control required">
                </div>
                <div class="form-group">
                    <label for="telephone">Telephone</label>
                    <input type="text" name="telephone" id="telephone" class="form-control">
                </div>
                <div class="form-group">
                    <label for="additional_message_label">Your Review</label>
                    <textarea name="additional_message" id="additional_message_label" class="form-control" style="height:120px;" onkeyup="getVals(this, 'additional_message');"></textarea>
                </div>
            </div>
        </div>
        <div id="bottom-wizard">
            <button type="button" name="backward" class="backward">Prev</button>
            <button type="button" name="forward" class="forward">Next</button>
            <button type="submit" name="process" class="submit">Submit</button>
        </div>
</form>

And following is my PHP.

<?php
    $mail = $_POST['email'];
    $to = "<sample@gmail.com>";/* RECEIVING EMAIL HERE */
    $subject = "Survey from DOMAIN NAME";
    $headers = 'From: Survey from DOMAIN NAME<noreply@gmail.com>' . "\r\n" . 'Reply-To: <noreply@yourdomain.com>';
    $message = "DETAILS\n";
    $message .= "\nA. How was the service provided? " . $_POST['question_1']. "\n";

    if( isset( $_POST['additional_message'] ) && $_POST['additional_message']) {
        $message .= "\nYour Review: " . $_POST['additional_message']. "\n";
    }
    $message .= "\nB. Would you reccomend our company? " . $_POST['question_2']. "\n";
        if( isset( $_POST['additional_message_2'] ) && $_POST['additional_message_2']) {
            $message .= "\nAdditional Message: " . $_POST['additional_message_2']. "\n";
        }
    $message .= "\nC. How did you hear about us?\n";
        foreach($_POST['question_3'] as $value) 
            { 
                $message .=   "- " .  trim(stripslashes($value)) . "\n"; 
            };
    $message .= "\nFirst name: " . $_POST['firstname'];
    $message .= "\nLast name: " . $_POST['lastname'];
    $message .= "\nEmail: " . $_POST['email'];
        if( isset( $_POST['telephone'] ) && $_POST['telephone']) {
            $message .= "\nTelephone: " . $_POST['telephone'];
        }
        if( isset( $_POST['age'] ) && $_POST['age']) {
            $message .= "\nAge: " . $_POST['age'];
    }

    $message .= "\nGender: " . $_POST['gender'];
    $message .= "\nTerms and conditions accepted: " . $_POST['terms']. "\n";

    //Receive Variable
    $sentOk = mail($to,$subject,$message,$headers);
                        
    //Confirmation page
    $user = "$mail";
    $usersubject = "Thank You";
    $userheaders = "From: sample@gmail.com\n";

    //Confirmation page WITH  SUMMARY
    $usermessage = "Thank you for your time. Your request is successfully submitted. We will reply shortly.\n\nBELOW A SUMMARY\n\n$message"; 
    mail($user,$usersubject,$usermessage,$userheaders);
?>

So when I submit the form, confirmation page successfully shows up. But when I check my mailbox, emails is not coming in.

Do I have to use Mailchimp to create this? Or are there any way I can send this form to email?

Thank you.