Start the outline of your argument essay by thinking about how you will introduce the topic to the reader. Without an introduction, the topic is lost. Clearly state the intention. Sometimes starting the essay by putting the point you wish to prove in quotation marks is an easy way to get going.
Make sure you don't start trying to explain why you believe in your point in this part, but rather use this section to show how you will prove the point. You can emphasize certain past debates on the topic, or simply discuss why the point you are trying to make is relevant.
Present the arguments. This is called the development of the essay. It is clear that, to draw out an argument essay, you need to have arguments where you prove the initial statement. These are drawn from your own experience, from previously read articles and books, other people's ideas, etc. Most commonly, two or three arguments are used in these sorts of essays. Remember that the arguments must be focused on proving why the statement is true, and they can't deviate too much from the topic itself.
More specifically, an argument is composed of certain facts, or a single fact, that point to the truthfulness of the writer's idea. You can start each argument with propositions of time, such as: firstly, secondly, thirdly, lastly, to start with, etc.
Complete the essay with the conclusion. This part usually reiterates the statement and restates the principal ideas of the individual arguments -- for example, "So, x is true because, as I have shown, y and z are also true." This is to reinforce your point and really bring it to the reader's attention.