A bit about my perspective. I am very non-confrontational. I hardly ever argue offline, and not so much online either. Although I have a lot of persuasive writing, so that's something. Also note that I've fully swallowed a particular piece of skeptical ideology: I believe in only developing argumentative skills that are more effective when deployed by the correct side than when deployed by the incorrect side.
90% of arguments are crap.
Sturgeon's law says 90% of everything is crap, and that applies to arguments as well. It follows that you are justified in being selective about which arguments to participate in.
It's worth pointing out that the most common failure mode is when arguments are not arguments at all. Sometimes it's a different kind of conflict entirely. Sometimes, people just explain their own opinion, and fail to offer any arguments for it. I wish people wouldn't try to pass off assertions as arguments, but on the other hand it seems like people don't spend enough time on assertions. I regularly get commenters who are unable to explain their opinion, and unable to understand mine. Obviously that stuff needs to be worked out before any productive argument occurs.
Argue from shared beliefs.
It's an Objectivist fantasy that arguments are built upwards from fundamental assumptions about reality. Human knowledge is more like a web of interconnected facts and opinions. You should think about who your audience is (whether it's the person you're arguing with, or bystanders), and either find out or guess what your shared beliefs are. Travel along the web from these shared beliefs towards the main subject of your argument. If that's too far to travel, argue about something else.
Be tactical, but honest.
Tactical arguments are not synonymous with dishonesty. For example, if your opponent believes A, and you disagree with A, you can still use arguments like, "If A, then B. So you should agree with me on B, if not A." For instance, sometimes I concern troll people (ie give advice to people I disagree with), and this is basically an attempt to argue based on the opponent's assumptions rather than my own. Whenever I do this, I am open about my role as concern troll.
Don't tell your opponents what they believe.
If you happen to be wrong, that's the quickest route to loserville. If you're right, your opponent may decide that they don't believe it after all, and whoops you're in loserville again. We do need to make guesses about our opponents (especially if they're no good at explaining themselves), but try not to be too brazen about it. If necessary, clarify that you are guessing their opinion, and ask if your guess is correct.
Treat scientific evidence as a trump card.
If the opponent presents a study, and you don't have the time or desire to check it, consider folding. If you do have the time and desire, check a) whether the study really says and implies what is claimed, b) whether the paper has questionable philosophical assumptions or methodological issues, c) whether the paper is authoritative, and d) whether there are superior studies out there.
People see themselves as essentially good.
This is such a strong cognitive bias, that it's better to work around it than to counter it, at least within any given argument. What people most need to hear is that they would not be bad people if they decided to change their minds. And this is often true; people make mistakes, and do bad things without being bad people.
People won't change their minds in front of you.
This is another strong cognitive bias that is better to work around. What people most need to hear is that if they change their mind about one particular argument for their beliefs, that doesn't mean they need to change their mind about the entire belief. This allows people to change their minds without changing their minds.
Tell your opponent something they didn't know.One acceptable moment for people to change their minds is when they receive new information. It's a plausible excuse for why they were previously wrong--they were simply unaware of a relevant fact.
Don't gloat when people change their mind.
Another thing is that people will sometimes change their mind when they think you're not looking. For instance, "shifting the goalpost", when it's not employed disingenuously, is just that. I do not recommend discouraging shifting the goalpost unless you think your opponent is being disingenuous.
Consider losing the argument.
Even if you don't really think you lost. There is some underrated power in being the loser: the loser chooses when to leave. You can get out of an argument that is either not fun or not productive.
And of course sometimes you really will be wrong. In those cases, you are probably affected by the bias where you don't want to change your mind in front of people. Losing even when you don't think you lost is a good way to give yourself the opportunity to change your mind--or not.