tl;dr a simple guide to setting up a LAMP stack on an existing Ubuntu server.

For my purposes I'm referring to LAMP as:
Linux (Ubuntu LTS): Operating system
Apache: webserver
MySQL: database
PHP: scripting language

Linux:
see alternate post

Apache:
Install Apache from the Ubuntu repositories

$ sudo apt-get install apache2

Configure Apache to start automatically

$ sudo systemctl enable apache2
$ sudo systemctl start apache2
$ sudo systemctl status apache2

Test Apache by opening the server IP address in a web-browser
server_ip
If everything went as expected you should see a generic page describing the Apache install and configuration files.

MySQL:
Install MySQL and set the root password

$ sudo apt-get install mysql-server mysql-client

Check the status of the service

$ sudo systemctl status mysql

Clean out some insecure default permissions

$ mysql_secure_installation


PHP:
Update the repositories, then install PHP and associated packages

$ sudo apt-get update
$ sudo apt-get install php libapache2-mod-php php-mcrypt php-mysql

Check that everything installed correctly by testing the current version of php that is installed (a version number should be displayed).

$ php -v

Once satisfied that everything has installed correctly, lets create a test page to confirm PHP is playing nice with Apache.

$ sudo nano /var/www/html/test.php

Enter the following code and save
<?php phpinfo(); ?>
Save the file, then restart Apache

$ sudo systemctl restart apache2

Navigate to the page we just created
server_ip/test.php
If everything works as expected we should see some details on the PHP installation.
Once satisfied everything is working as expected, remove the page just created.

$ sudo rm /var/www/html/test.php


Congratulations, you've now got a server setup and a LAMP stack installed.