In general, if you're arguing against someone, you have to actually know what you're actually on about, otherwise you're effectively trolling.
If you're arguing against someone of a different worldview, you have to understand the other worldview. Remember that the assumptions are different. Even I do this. However, also remember that truth is objective, and that there's a possibility that you may both be wrong, let alone yourself.
Speaking of objective truth, if you see something contradictory to your experience, that's a good time to say "no, that's not the case."
You are not necessarily arguing on your grounds. Nor are you necessarily arguing on the opponent's grounds. You are arguing on the grounds of logic. Logic is a good thing, but as we're all human it's essentially guaranteed that you'll botch up your logic every once in a while; hence why the argument is not on your own grounds, as you can't change logic (it's a discovered thing, not an invented thing, but I guess some may have had it wrong in the process - this is purely a (calculated) guess, though).
Context is your friend unless you're a Youtube atheist. Why do I put it this way? Because Youtube is a "brilliant" place to whinge about anything and everything, and the average whinger, if religion is important, is probably an atheist. They're probably a "liberal", too, mostly because it just sounds good. This is an assumption, so if you're going to prove me wrong, I'd like some statistics. And the real thing, not just the inverse function analysis.
This is not an attack on all atheists on Youtube. Mostly just those who leave rather lame comments which have no citations whatsoever (and are in a similar key to this very sentence). A teacher told me in high school that "empty vessels make the most noise".
As Mark Twain allegedly said, there are three kinds of lies: lies, damned lies, and statistics. Well, someone said it before me, because I read it somewhere!
Remind yourself of this phrase, as is:
Your opponent may make some rather strong claims [citation needed].
Seriously, if your opponent makes a "rather strong claim", ask where the evidence is. However, if you know what is talked about, and you know the evidence is contrary to what is spoken, you can point this out with the "no, that's not the case" manouver.
If you have to point out how rational you are, your motive for arguing is probably irrational.
Avoid "textbook liberal". Maybe you're right but it makes you sound like a bigot. This is what I would call "textbook liberal" (this is just an example, don't shoot me down for it):
Atheists are very arrogant. Anyone can see that they are in the wrong. You can't say that they play a necessary part in our society. There is absolutely no reason to listen to them.
I'm trying to remember this letter written in the paper by someone who I think is someone I know and it was embarrasingly "textbook liberal". Hint: the keywords are "anyone", "absolutely", and "you can't". (The letter was about that MP who was caught for spending a lot of taxpayer money on porn & stuff semi-recently in New Zealand, I forget his name but I believe he was a Labour MP.)
You might have quite a strong opinion. It's a good sign of your ability to stand for something, but it can also be quite dangerous. I look at Peter Hitchens' blog, and it's clear that there are opinions, although sometimes I shudder at some of the things which are said.
To look at his blog I've had to open up Lynx as for some weird reason the text doesn't appear to show up in Firefox (I've tried disabling adblockplus and I've got noscript in place, but to no avail). This is lynx, by the way (I accidentally took a screenshot without the window decorations, and no, this isn't on Linux, it's on FreeBSD):

This is probably a wording issue, but this is an example (and the only one I can see ATM):
The average squirrel could see that if you treat criminals of any kind with sympathy and kindness, you'll get more criminals.
I think I understand the thinking behind it, but that it was worded badly. A more appropriate way to put it would be that if you encourage people in their crime instead of trying to get them out, then you'll end up with more criminals, but this isn't what's written so I don't explicitly know what's meant.
However, I think that treating criminals with "tough love", I guess, would help cut down crime, contrary to what is explicitly written.
I suggest you read the blog overall, though, as yes, occasionally journalists get things wrong.
There seems to be an issue with people you don't really know and/or understand, the issue being that you don't know and/or understand them. I knocked homosexuals until I met at least one (well, bisexual anyway) on the internet. I was somewhat afraid of Muslims until I joined in a discussion with one about faith. It's at that point that you realise that there are wonderful people out there who are lost.
Having said that, I don't want to wait until tomorrow to go to bed, so good night and may God always be in reach.