How to contribute to wikipedia

By now, wikipedia has become quite famous. We also now, that the all the content there is contributed by users for free... and we also thought about writing articles for it. But how to start? What do I have to do exactly and what mistakes could I avoid as a beginner? Lifehacker has a great tutorial on that.
And by the way... you are already a big help if you fix typos in existing articles or add some paragraphs.