How do I co-author my first paper?

You don't first co-author a paper, you first collaborate on a research effort. Then, if said research effort leads to interesting results, you co-author a paper. So you need to look for someone doing interesting research that you think you can contribute to. Talk to professors in your department and ask about the research they and their students are doing.

Of course, since co-authoring a paper is important to you, when you find the opportunity to be part of a research project you should ask the professor: "Do you think this is likely to lead to a paper before I graduate?" Nobody can promise you a paper (even if you do everything right, the nature of research is that we don't know everything in advance). But he/she will probably have some intuition about it. You should also ask what you specifically need to do to merit co-authorship credit on the paper that comes out of the research.

This depends on the specifics of your institution, of course, but at my university, there would be two options:

• A very good student project: Start a student project, and put twice the time and energy into it. The important thing to do here, is to tell the project supervisor that your aim is to turn it into a publication, and you're willing to put the extra work in. It may seem a little emberrasing to be so forward about your ambitions, but it's a very good ambition to have, and your supervisor should respond positively. The point is, if you don't tell them, there's no guarantee that they'll decide to turn it into a publication, no matter how good it is. It simply may never occur to them. They may also suggest ways of lowering the bar a little, like making it a conference paper, and sending it to a national conference rather than an international one.
• Join an existing research effort: This is trickier to do, but once you get accepted, the chances of success are higher. First, you have to make sure you have a good reputation with your teachers, and you have to be lucky enough to get a part-time job assisting in the research. Whether such jobs exists depends on your institution again. I can't think of anyone who would unload a substantial part of his own research work on an unpaid student. Again, whatever opportunity you find, make sure to communicate your ambitions (without demanding anything), so people can help you fulfill them, or tell you that it's unlikely to happen.

There's no surefire way to achieve this, unfortunately. Just make sure that you tell people that you're trying to get a publication, and whenever you think a project might yield publishable results, put in the extra hours. Even if you fail, your efforts will be good exercise for the next attempt. Also remember to pick your battles: if you try to make every student project into a paper you will probably spread yourself too thin.