What is the fastest way to build a site together using PHP, JQuery and MySQL?

What you are asking is actually called CMS There are few excellent open source ones like drupal, wordpress. I persoanly prefer wordpress install XAMPP http://www.apachefriends.org/en/xampp.html And then Wordpress  http://wordpress.org/ Both of the installs should take around 10 minutes and you have a running system.

To be simple blog/static website - wordpress Typical CMS - Drupal Complex Application - PHP framework like codeigniter Simple application with more work on Front End - Core PHP, just use lamp or wamp and proceed!

Depends on the "site". If you are building something like a blog, fastest way would be using an open source application (i.e. using WordPress for blog) for your purpose. If you are a PHP web developer, "fastest" way would be using a framework with code generator. (i.e. Yii Framework)

First, the server. Server is APACHE! If you don't want to have troubles installing everything on your mac, I would recommend you to use Xampp - http://www.apachefriends.org/en/xampp.html Xampp will install Apache, PHP and MySql. So, you are ready to code! For coding, I'm using, and loving, the Komodo Edit IDE - http://www.activestate.com/komodo-edit/downloads With Komodo you can write codes on JavaScript, Perl, PHP, HTML, CSS, Python, Ruby and Tcl. Talking about browsers I'd recommend using Firefox or Chrome, since both have a extension called FireBug, wich helps you debug Html, Css and Javascript. At last, I would like to suggest you to learn/use the Smarty template engine - http://smarty.net It's a small engine written on PHP to help you on the MVC design pattern.

I would suggest starting with Yii, http://www.yiiframework.com/features/ Yii is integrated from ground-up with jquery and is a fantastic PHP framework.

First you have to know what you want. Wordpress can solve blogging, news sites, or act like a little cms. If you need something more complex, try Drupal. You can buy or download free templates for those, and there is a lot of plugin and modules to expand functionality, including jquery fx. If is something More complex, you can't do it fast. I also recommend Code Igniter as framework for complex applications (MVC pattern, non special server requests, no need to use command line)