Check if they are logged in and then display only the buttons you want?
Where are you lost?

Basically you will create a session object when a user logins. If a session is created, then do not display the login button.

You need to create a session when the user logs in as in...
<?php
session_start();
if(!isset($_SESSION['user_id'])) {
if(isset($_COOKIE['user_id'])) {
$_SESSION['user_id'] = $_COOKIE['user_id'];
}
}
?>
This code above will check to see if a session exists for the user. A session lasts until the browser is closed out. So it will also check to see if a cookie was set as well (google 'setcookie php' for more details.

Now, anytime we want to check if a user is logged in, we simply call the $_SESSION['user_id'] variable.

Now we need to apply that to your menu. So basically, we'll check the variable before writing the login/logout menu.
<?php
session_start(); // this will keep the user's session alive
echo '<ul id="login_menu">';
if(isset($_SESSION['user_id'])) {
echo '<li><a href="logout.php">LOGOUT</a></li>';
} else {
echo '<li><a href="login.php">LOGIN</a></li>';
echo '<li><a href="register.php">REGISTER</a></li>';
}
echo '</ul>';
?>
You can make adjustments to the structure as you like, but the if statement is the key. If the user is logged in (isset) then you to display the appropriate menu. Otherwise, the other menu is displayed.

If you need more help, please let me know...

You can modify and try to change your PHP script. But for the simple way and get perfect result, you can install paid widget plugin that you can found through internet. You can choose the kind of widget that you wish.

You need to write code that verifies whether or not someoene is logged in on each page load. If they are, then load the logged in version of the page. If not, then bump them back to the login page. Then just update your logged in page template to include a logout button instead of login.

Maybe you should use a solution like PHP User Class - php user class

Otherwise if you want to fix what you have, you will need to play with $_SESSION and probably a database in order to keep track of your users.

Hope that helps

try this...

<form action="/login" method="post">
<p>
<label for="email">Email address:</label>
<input id="email" type="text" name="email" />
</p>
<p>
<label for="password">Password:</label>
<input id="password" type="password" name="password" />
</p>
<p>
<input type="submit" name="login" value="Login" />
</p>
</form>