PHP is just a scripting language so all you need is a text editor. Preferably one with syntax highlighting; I tend to use
ConTEXT. That, and a web browser pointing to a tutorial and/or the PHP website.
You'll probably want to get a local webserver set up for testing. I use
WampServer, which is Apache, PHP and MySQL for Windows. Then, with a little configuring, you can have a test version of the site on a local disk that you can view in a web browser, rather than having to upload to test changes. This is assuming the site's running on similar software, of course.
Synchronising your local disk copy with the site is another issue. I use rsync but it's quite hard to set up.
A word of warning: PHP code is easy to write, but hard to write well. 99% of what you find online, posted as "tutorials" or "tips", is in fact rubbish.
If you do ever decide to rewrite the site from static HTML to be DB-driven, a framework like Ruby on Rails (probably not supported by your host) or CakePHP (PHP clone) is probably a god start. But that's probably at least a few years off.