The first step is to determine your position. Decide what you want to argue about, and what position you will take on the issue.
The next step to argue effectively is to come up with support for your stance. It's always a good idea to have at least 3 or 4 strong and solid support points for your argument. You can make things up if you have to.
Organize your ideas into a few categories, it's best to take make a logical argument, and supplement it with emotion. Don't focus your entire argument on either logic or emotion.
Anyone who knows how to argue effectively will know that you need to refute the other person's claims! Say why you think the other person and their reasoning is wrong. Address each claim individually.
Use proper language when describing your position. Words like therefore, vis-Ã -vis, and ergo make you sound intelligent and organized in your speaking. These words also help your audience follow your line of thinking.
Finally, be able to change and manipulate your argument. If you discover you are wrong in the middle of your argument, don't be afraid to adapt a little to compensate. The key is to give a little compromise, but still come ahead in the end.