What are the key programming languages that front-end web developers should know? How about for back-end web developers?

A lot of languages share enough commonality that you'll be able to pick up new ones as you progress in your career. Don't tie yourself to any one (or set) of languages.

That said:

====Frontend====
Javascript, HTML, CSS. Mostly because web pages are built using these three.

First and foremost javascript. Also, you need to have a good working knowledge of HTML (not really a 'programming language'), and you need to be able to at least fight with CSS enough to make it do stuff. Ideally you'd be good at making clean CSS, but, well... it's CSS, so if you do actually become really good at it (especially the odd parts), people will revere you as some kind of god and/or evil wizard (only half-joking here).

Then, if you want to look at the mobile end of frontend, you're talking Java, ObjectiveC, Swift (the latter two only for iOS, the former for Android).

Some places will do web stuff in PHP, Java, etc.

====Backend (web) ====
Take your pick. Java, C++ are still heavily used. On the Microsoft .NET side there's C#. But there's a long list that can be used in backend: java, c, c++, c#, javascript (via node.js), python, ruby, go, perl (pls don't), pascal, lisp, haskell, etc etc etc.

It will also be useful to have a good command of SQL type stuff (again, like HTML/CSS, this isn't really a 'programming language').

The distinction between frontend and backend programming is temporary. Don't get overly distracted by it. The technology will change at least six times, maybe more, during your career.

Python is a good place to get started programming. You need to understand data. SQL is a good place to start there. Much of the world's data is stored behind SQL-based interfaces these days. You'll need to know Javascript, for sure.

As you think about learning to "program" you might want to read this article. http://www.posteriorscience.net/?p=206 When I started programming, we built a lot of stuff from scratch. Now, we use vast runtime libraries and frameworks, and piece together the functions we need. Neither approach is right, they're just different.

Also check out http://www.w3schools.com/ which have great tutorials on a number of programming languages from Java, SQL, HTML etc. It will give you a good background on the multiple languages and technologies out there. You don't need to install anything on your machine either as it has compilers and scratchpads built in.

Do however look at your documentation and writing skills too. There are a lot of programmers that don't document correctly and when someone needs to review their code it can be difficult, so get into the habit of documentation and commenting your scripts.

I can't speak much to backend but for front end it's important you have a strong foundation in CSS and JavaScript.

Once you have that you can start looking into things like LESS and Sass or Grunt and Gulp.

LESS and Sass are CSS preprocessors and make a huge difference in the development process.

Grunt and Gulp are JS based and built in node and are used for automation. They can be used for a wide range of tasks.

It's very important you learn plain (vanilla) JavaScript and not just a library like jQuery. Having a strong understanding of JavaScript will help you adapt to the newer technologies.