How to use PDO connection in other classes?

Question

I think I've a problem in understanding how OOP works. I already changed the code that it works, but it isn't the propper way I think. Following scenario (No, I'm not creating a userlogin by myself, its really just for local dev. to understand OOP better):

I've a database.php file:

class Database {

    /* Properties */
    private $conn;
    private $dsn = 'mysql:dbname=test;host=127.0.0.1';
    private $user = 'root';
    private $password = '';

    /* Creates database connection */
    public function __construct() {
        try {
            $this->conn = new PDO($this->dsn, $this->user, $this->password);
        } catch (PDOException $e) {
            print "Error!: " . $e->getMessage() . "";
            die();
        }
        return $this->conn;
    }
}

So in this class I'm creating a database connection and I return the connection (object?)

Then I have a second class, the famous User class (actually I'm not using autoload, but I know about it):

include "database.php";

class User {
    /* Properties */
    private $conn;

    /* Get database access */
    public function __construct() {
        $this->conn = new Database();
    }

    /* Login a user */
    public function login() {
        $stmt = $this->conn->prepare("SELECT username, usermail FROM user");
        if($stmt->execute()) {
            while($rows = $stmt->fetch()) {
                $fetch[] = $rows;
            }
            return $fetch;
        }
        else {
            return false;
        }
    }
}

So thatare my two classes. Nothing big, as you see. Now, don't get confued about the function name login - Actually I just try to select some usernames and usermails from database and displaying them. I try to achieve this by:

$user = new User();
$list = $user->login();

foreach($list as $test) {
    echo $test["username"];
}

And here comes the problem. When I execute this code, I get the following error message:

Uncaught Error: Call to undefined method Database::prepare()

And I'm not sure that I really understand what causes this error.

The code works well when I change the following things:

Change $conn in database.php to public instead of private (I think thats bad...? But when its private, I can only execute querys inside of the Database class, I'm right? So should I put all these querys in the Database class? I think that's bad, because in a big project it will get become really big..)

And the second change I've to do is: Change $this->conn->prepare to $this->conn->conn->prepare in the user.php file. And here I've really no Idea why.

I mean, in the constructor of the user.php I've a $this->conn = new Database() and since new Database will return me the connection object from DB class, I really don't know why there have to be a second conn->