I taught argumentative writing at the college level. If I were to boil it down to a few basics, here's what I'd say.
1) Be clear, in every sense of the word. Have a clear idea of what your point is, have a clear sense of why your position is better than the others you've considered, and present your position in writing that is as clear as you can make it. Clear ideas, clear structure at the paragraph level, and clear grammar. Poor grammar and/or spelling make you lose authority with the reader, thus weakening your ability to persuade.
2) Say what you mean and mean what you say. Be sure that you are making the points you intend to make and you are not accidentally saying something else. (Clarity, revisited...) Also, make points that matter to you; you'll generally argue better if you care about what you're saying.
3) Don't get hung up on the 5-paragraph theme model. It is a decent starting point, in that it asks for an introductory paragraph, some evidence in the form of supporting paragraphs, and a conclusion. But there's nothing magic about 5 paragraphs. If you have two or four very strong reasons for your position, use that number. You'd be amazed how many really bad third reasons I've read precisely because people didn't really *have* third reasons, but invented lame things to fit the model. Awful!
4) Shouting down or insulting your opposition doesn't constitute winning the argument. Truly persuasive argumentation takes into account where the audience is coming from, addresses those views respectfully, and offers evidence to show why this other position is better. It is best to assume that the opposition is not evil, stupid, or part of some grand conspiracy. The sayings "You catch more flies with honey than with vinegar" and "Don't preach to the choir" are good thoughts here.
5) Reasons need to support the thesis statement and evidence needs to support the reasons. Don't throw something in because it sounds good. If it's irrelevant, leave it out. An argumentative essay is like a building: the evidence is the foundation, and if the evidence doesn't support the thesis, the building won't stand up.
6) Edit at the structural level before you proofread. If you don't have a thesis statement, it won't matter if you missed a comma somewhere.
7) Introductions are your chance to set your argument up in a positive light. The funnel structure is a good one for an intro: offer useful background on the issue you'll be discussing, then move toward the thesis statement at the end of the intro. The basic movement is from general to specific. There are other formats, but it's probably the simplest one to master. Just don't get too general--avoid "Since the beginning of time..." openings. Very few things have been happening since the beginning of time, and even if your issue is one of those things, is that actually relevant? Probably not!
8) Conclusions are your chance to make your point "stick." Simply restating your argument in summary form is better than nothing, but if you can go a step further and give a "so what?" for the reader, a reason why this should matter or how it can be applied in real life, that's even better.
HTH!
