The above answers are all very valid and in the end, they represent your target solution.
In the meantime, you may already do a lot for your website, even with a gradual migration to those practices.
In order to do so, I suggest you to install PHPUnit (or whatever Unit comes with the web languages you use). There are also "graphical" versions of it, like VisualPHPUnit, if that's more of your taste.
These tools are not the permanent solution. You should actually aim adding them to your permanent solution, that is setting up development server etc. However, even as interim solution they help you reach a fairly stable degree of quality for your software components and avoid 80-90% of the surprises that come with coding on a live server.
You can develop your code in a separate directory and test it before you move it into production. You can create mock objects which your code under test may freely interact with, without fear of repercussions. Your tests may load their own alternate configuration so they work on a second, copy database. Moving even further, you may include your website into tests itself. There are several applications like Selenium that allow you both to automate and test your production website, so that you can reliably know that your latest changes did not negatively affect your website semantics.
In short, while you should certainly aim at getting a proper development environment running, you can do something very good even today, with few hours of study.