It very much depends on who I am talking to. Most people are not interested in politics or economics. For these people, I try to use short, utilitarian arguments to grab attention (pointing out the benefits of free-market education, healthcare, police, courts, etc). If the person shows an interest I will bring in the ethical side, which is a form of argument people are not so used to, but which can be compelling. No other political economists can use coherent ethical arguments. Like if somebody objects to me advocating abolishing welfare, I will say “so you think it’s OK for the government to steal money from everybody to give it to people who don’t have a job?”
Talking about current economic events is also a good way of starting a conversation, and people are becoming increasingly interested in what I have to say. I am getting good at explaining the Austrian business cycle theory to a layman in under one minute.
If I’m speaking to someone with an economics degree (who invariably have not heard of the Austrian school, because their professors never mentioned them, so they must be wrong!), I talk to them about fundamentals and the epistemological basis of economics. How can they justify viewing human beings as purely subject to physical laws (i.e. lacking choice) - the prerequisite for empirical science? How is the objective theory of value even remotely true? How can inflation and the business cycle be caused by anything other than changes in the money supply?
If I’m speaking to a loyal political party member, I’ll point out the inadequacy of the Left-Right paradigm and how the major parties actually represent merely different forms of socialism. I’ll point out how any government is always bad - economically, ethically, and because they always tend towards totalitarianism. I’ll oppose anything anyone says in support of government action.
I’ll usually direct people to mises.org or my own website where I have created a short course in economics (here). I quite often send a follow-up email after discussing something with someone, with links, and invite them to have an email discussion with me so that they (and I) can more fully think through the arguments.
I find it easier to convert “athiests” (people with no economics degree nor interest in politics) but they tend to be less pro-active once converted on any one issue to learn more about the whole libertarian viewpoint. Keynesians and those who support a mainstream political party are far harder to convert, but do seem to do more research after they have been “hooked” - making months of endless debating pay off.
Anyway, I ramble - hope this helps!