<?php
if(isset($_POST['register'])){
$first_name = $_POST['first_name'];
$last_name = $_POST['last_name'];
$email = $_POST['email'];
$password = $_POST['password'];

$userdata = array(
    'user_login' => $email,
    'user_pass' => $password,
    'user_email' => $email,
    'user_nicename' => $first_name,
    'display_name' => $first_name . ' ' . $last_name,
    'first_name' => $first_name,
    'last_name' => $last_name
);

$new_user = wp_insert_user($userdata);

$cred_array = array(
    'user_login' => $email,
    'user_password' => $password,
    'remember' => true
);
$login_user = wp_signon($cred_array);
if ( is_wp_error($login_user) ){
    echo $login_user->get_error_message();
}

}

?>