Start on the client side with JavaScript (JS). That's the only language supported by all the major browsers. Get the language basics down, and then start learning about the Document Object Model (DOM).
That brings up an important point: Even if you're a developer, rather than a designer, you'll need to know HTML for markup and CSS for style/presentation. Those aren't programming languages, but they are languages you'll use a lot. The DOM is JavaScript's representation of the HTML and CSS that's been downloaded and parsed by the browser. HTML is also where you connect your JS code to events and on-screen controls.
On the server side, there are more choices. I'd suggest you start with PHP as the most common back-end language. This is available on just about any cheap or free shared hosting plan. Other languages generally require some extra server side admin support, but you'll find Python, JS, Java (the real language, no relation to JS), C#, VB, Ruby and others in use on real websites.
You'll need to know about relational databases (the only kind of databases that most devs know about these days) and learn SQL (Structured Query Language). Some people consider this a programming language, some don't. Doesn't matter, really...you still need to know it.
If you're maintaining a website on the server side, it helps to know at least the basics of command line interaction with the OS. For a web server, that's usually Linux and the command-line shell will usually be bash. If your development computer at home is a Windows PC, you can get a feel for this by installing Cygwin and using it's bash shell. On a Mac or Linux, you'll have bash built-in as the default shell whenever you start at terminal window.