Linux variant: depending on the distribution you use, there are (most certainly) prepared packages (e.g. Debian, Ubuntu) to install, if you use a more complicated distro like gentoo, you have to download and compile the source packages (check Google for tutorials on that).
Windows variant 1: you can employ a prepared WAMPP package (apache, mysql, php, pear/pecl) - find it here: www.apachefriends.org
Windows variant 2: you can install all desired services manually (check google for tutorials on windows apache iis)
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One of the easiest ways I have found to get MySQL running on a LAMP server is to actually install phpmyadmin through the terminal. This will load everything you need to get LAMP running, Apache, PHP and MySQL. You may need to install mysql first before running this but I have found it to be the easiest way to get all of it running on a server. Many hardcore admins may tell you to never use apt-get and install everything from source and compile it yourself but if you are a novice, apt-get is a good way to go. The commands for loading mysql are "sudo apt-get install mysql-server; sudo apt-get install mysql-client;" depending on your version of Linux. If you're on Windows, give up and start a linux box.