Let's start with the first tactic conservatives use in argument: oversimplification. Usually, this involves many broad topics that conservatives are not intimately familiar with, either because they are uneducated or indifferent to them, yet they feel inclined to express their opinion on them. These things include, but are not limited to: history (conservatives are especially uneducated on American history), American politics, international relations, race relations, public policy, urban development, education, finance, economics, sciences (especially natural science), logic, philosophy, and math. In fact, it seems to be the case that unless the argument is heading in the direction of NASCAR or drinking copious amounts of Budweiser, conservatives will have little to no education on the topic. Here is a small example of conservative oversimplification and a simple analysis:
“9/11 happened because all Muslims are terrorists, because Islam is a terrorist religion that hates everyone and they hate the American way of life.”
Perhaps I should classify this classic conservative line as an aphorism; while it does include oversimplification, it goes back to a general statement that is taken as a maxim because no contrary evidence exists as conservatives are too busy to read books and news articles. (you'd be surprised how much time hunting and painting a confederate flag on your 1972 El Camino takes.)
Usually, after this gem is dropped, an educated leftist will scratch his or her head, nonplussed at the sheer stupidity that just saturated the room. After searching for a clause or phrase that could possibly ease the sharp pain that has grown in his or her pre-frontal cortex, the leftist usually regresses into a state of defeat, realizing that it doesn't matter how many facts or how much empirical evidence he or she can provide to said conservative – they're just too dumb (or too ignorant, racist, etc.) to comprehend world events, and would rather settle with a simple phrase or statement that sums up the often complex problems of international affairs because thinking too much hurts their heads.
Usually, the leftist (remember, leftist is distinguished from liberal) will do what he or she normally does – provide the conservative with the real reason 9/11 happened. I'll do it in dialogue format, just because it'll be more fun:
Conservative: All Muslims are terrorists.
Leftist: Uhh that's like me saying that all Christians are..
Conservative: America is a Christian nation. Are you about to call an American Christian a terrorist?
Leftist: No, but they do some shitty things to other nations..
Conservative: What, do you hate this country? This country was built on Christianity and we're the greatest country in the world -- how can we be wrong?
Leftist: But at what expense does our “greatness” come? Not to mention, if this country is so 'great,' why has the unemployment rate increased steadily since the initiation of Reagan type policies? Why is more than half of the wealth concentrated in the hands of 20% of Americans?
Conservative: What do you mean expense? We don't do anything wrong; other people want to kill us for no reason – because they're terrorists and because Islam is a terrorist religion.
Leftist: Again, how are you oversimplifying this and categorizing 25% of the world's population as de facto terrorists because they're part of a religion? I can make that case about Pat Robertson and Christians in America, and I certainly can make that case with Zionists who call for the destruction of the Palestinian people and support the apartheid policies of the state of Israel.
Conservative: Well there's too much terrorism in Israel. 9/11. What about 9/11? Have you forgotten about 9/11? They attacked us. We didn't do anything, and they killed innocent civilians who didn't do anything.
Leftist: Okay, so you're saying that “individuals who kill or perpetuate wanton massacre of civilians are terrorists?”
Conservative: Yeah, like the radical Muslims on 9/11.
Leftist: By that definition then, almost every United States president is a 'terrorist.' Bush sent drones to blow up hospitals and schools in Iraq, who were civilians, Clinton killed plenty of civilians in Kosovo on the same day as Columbine, and certainly Reagan, pursuant to the Domino Theory, forcibly removed democratically elected leaders in South Ameri...
Conservative: What? If you hate America so much, why don't you get out of the country and move somewhere else?
Leftist: Because this is my country.
Conservative: Well why don't you move to Cuba or North Korea you communist? Huh? There's no democracy there you would hate it and come rushing back.
Leftist: Right, which is why I don't want to go there, I want to stay here I just want to make changes.
Conservative: We don't need changes we're the greatest country in the world. Okay I have to go; I'm on my way to Arizona to sign up for the neo-Nazi anti-immigrant rally.
That's more or less how conversations with conservatives go. It's frustrating because they're so uneducated that it would necessitate a few days straight to enlighten them to the facts, and you would not be able to go 15 minutes without being interrupted with patriotic rhetoric. In fact, I'm going to create a mathematical law to represent the relationship between conservative rhetoric and the use of fact(s):

Either way, that is just an elementary example of the oversimplification problem. A few of my other favorites (and the proper respective response when warranted):
“Saddam had nukes; it was either us or them.”
“Survival of the fittest applies to us, too.”
“People don't have money because they're lazy and they don't want to work, not because there are no opportunities.” Or because education is not fair for every individual; if you look at the difference between allocation of capital per student at two locations, say Red Bank, NJ compared to Elizabeth, NJ, I'm sure you'd quickly see how per student, Red Bank gets a lot more and thus gets a much better education.
“Illegal immigrants ruin hospitals by making them bankrupt.” Illegal immigrants are good for the economy, (source)
“if you don't support the troops, the terrorists win”
“we need guns to defend ourselves, and somehow the interpretation of the right to bear arms which in context was used to justify militia during the American revolution justifies me in getting an AK-47”
“if we start letting gays marry, it would be okay for people to marry animals”
“all those damned liberal hippie professors at universities are brainwashing our kids to follow their conspiracy liberal agenda with meaningless things like facts and statistics and perspective”
“there's no need to study philosophy because it won't help you get a job when you graduate”
Bulletproof logic.
What makes this task so difficult is that it takes a legitimate amount of emotional and intellectual capital to remain calm while attempting to enlighten the right to how things actually work. As seen in the oversimplification complex, it is difficult trying to explain anything to a conservative given their inability to comprehend complex issues. Moreover, the absence of an established education in most affairs makes this task exponentially more difficult. What is shameful, however, is that most of what conservatives say has political appeal, at least for conditioned Republicans like Palin et al. Since a majority of Americans don't really care about what's going on, they leave it up to these aphorisms and equivocations and so forth to propagate political opinion (this is why Obama's middle name and de facto association with Islam was so prominent during the election) rather than allowing factual discourse to take place. Instead of arguing, do something more meaningful and productive, like taking a nap.