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

see alternate post

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
If everything went as expected you should see a generic page describing the Apache install and configuration files.

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

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
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.