woocommerce change html structure of checkout page

Question

Currently trying to change the html structure of the checkout page in woocommerce. I was looking for the template file, so I could add a title here and there. Some re-ordering of the input fields, like e-mail address at the top. But I can't find the template or way how to do it.

I did find the checkout form php file, but that is not what I was looking for. The code of the file:

<?php
/**
 * Checkout Form
 *
 * This template can be overridden by copying it to yourtheme/woocommerce/checkout/form-checkout.php.
 *
 * HOWEVER, on occasion WooCommerce will need to update template files and you
 * (the theme developer) will need to copy the new files to your theme to
 * maintain compatibility. We try to do this as little as possible, but it does
 * happen. When this occurs the version of the template file will be bumped and
 * the readme will list any important changes.
 *
 * @see https://docs.woocommerce.com/document/template-structure/
 * @package WooCommerce\Templates
 * @version 3.5.0
 */

if ( ! defined( 'ABSPATH' ) ) {
    exit;
}

do_action( 'woocommerce_before_checkout_form', $checkout );

// If checkout registration is disabled and not logged in, the user cannot checkout.
if ( ! $checkout->is_registration_enabled() && $checkout->is_registration_required() && ! is_user_logged_in() ) {
    echo esc_html( apply_filters( 'woocommerce_checkout_must_be_logged_in_message', __( 'You must be logged in to checkout.', 'woocommerce' ) ) );
    return;
}

?>

<form name="checkout" method="post" class="checkout woocommerce-checkout" action="<?php echo esc_url( wc_get_checkout_url() ); ?>" enctype="multipart/form-data">

    <?php if ( $checkout->get_checkout_fields() ) : ?>

        <?php do_action( 'woocommerce_checkout_before_customer_details' ); ?>

        <div class="col2-set" id="customer_details">
            <div class="col-1">
                <?php do_action( 'woocommerce_checkout_billing' ); ?>
            </div>

            <div class="col-2">
                <?php do_action( 'woocommerce_checkout_shipping' ); ?>
            </div>
        </div>

        <?php do_action( 'woocommerce_checkout_after_customer_details' ); ?>

    <?php endif; ?>
    
    <?php do_action( 'woocommerce_checkout_before_order_review_heading' ); ?>
    
    <h3 id="order_review_heading"><?php esc_html_e( 'Your order', 'woocommerce' ); ?></h3>
    
    <?php do_action( 'woocommerce_checkout_before_order_review' ); ?>

    <div id="order_review" class="woocommerce-checkout-review-order">
        <?php do_action( 'woocommerce_checkout_order_review' ); ?>
    </div>

    <?php do_action( 'woocommerce_checkout_after_order_review' ); ?>

</form>

<?php do_action( 'woocommerce_after_checkout_form', $checkout ); ?>

So the result I want is to re-order the input fields (e-mail at top for example), adding some text here and there and it would be amazing to add some custom classes.

The only thing I found online is to add some filters/hooks into the functions.php. But it feels a little bit hacky to do that.

Edit / additional info:

Sorry if my question wasn't clear. I'm trying to achieve this layout:

<div class="column">
    <h2 class="title">Billing Details</h2>
    <?php if ( $checkout->get_checkout_fields() ) : ?>

    <?php do_action( 'woocommerce_checkout_before_customer_details' ); ?>

<h2> CONTACT INFO </h2>
        <div class="field">
          <label class="label">First Name</label>
          <div class="control">
            <input class="input" type="text" placeholder="First Name">
          </div>
        </div>
<h2> SHIPPING INFO </h2> 

        etc......

    <?php do_action( 'woocommerce_checkout_after_customer_details' ); ?>

    <?php endif; ?>
</div>

As you can see I want to re-order the field, but also adding some html tags in it for titles or additional texts. That's why I'm looking for changing the HTML structure/template, instead of the array order of the fields.