This post isn’t going to be about persuading people in the conventional political realm. For that, you should ask a politician. This post is about debating political ideas with people, whether in person or on the internet.
The first thing that you should know is that you’re unlikely to get someone to cede their position or their fundamental beliefs in the course of a discussion. People will tend to be less focused on whether or not their position is right and more on how they can show that they have the stronger position. Because they’re focused on that, rather than on figuring out what’s actually true or preferable, you’re unlikely to shift them much in the course of a discussion.
Emotions will also tend to run high in a debate. There are a couple ways to handle it depending on your objectives and the rules of conduct of a given discussion space. You can either be emotionally withdrawn, or you can use passionate expression to bowl over the other person’s position. What your personality is like and what the situation is should direct what method you choose to use.
What you can do is shift the beliefs of other people who might be watching the exchange. And you might be able to indirectly influence the beliefs of the person that you spoke with originally on a longer timeline than the course of a given discussion.
Knowing all this can save you a lot of frustration.