Identify the main point you want to make. Starting with your own set of beliefs, consider how you feel about the topic and where your passions lie. The best presented arguments are the ones that come from the heart.
Consider who your target audience is and what its current beliefs might be. You're not going to bring up the same points to a group from the National Rifle Association as you would to a group from People for the Ethical Treatment of Animals.
Once you've chosen your side, do the research that supports your point of view. Use the opinions of experts, bolstered by examples and statistics. Quotations, effectively placed, can also grab your opponent's attention and sway him to your ideas.
Don't ignore the other point of view. Instead, take a close look at the other side and challenge it at every turn. Be respectful in tone and demeanor, and rebut each point logically and concisely.
Whether you're presenting your case in a debate forum or on paper, being well organized is imperative if you want to keep the attention of your target audience. If the people you are trying to convince don't understand what you're trying to say, you've lost the argument before you've begun.
Grab your target audience from the beginning with a quote, a statistic, an anecdote or even a joke and you're more likely to keep their attention. Likewise, when you close, be certain to summarize in a way that reminds them of the most important reasons your idea is best.