What It's All About:
Someone just said something that you disagree with. There's no way that they could be right, so the trick is figuring out two things: why they're wrong, and how to best make this clear to everyone listening. Note that the first step is key! There's nothing more annoying than someone who always loudly insists that he is right, but can't ever provide any intelligent thinking to back it up. Don't be that guy.
Know Thy Enemy:
Before you can hope to refute someone's argument, you have to understand what they're actually saying. Sometimes this will be easy; sometimes it will be the most difficult thing in the world. In debate, the case statement is usually a good place to start, but don't be complacent. Hold them to a single, consistent position. Make them define any confusing terms. Listen and check that each of their arguments actually supports their case, as you understood it. There are two benefits here: you're better able to address what they're actually trying to argue, and they're not able to shift their argument out from under you. A lot of times, you can very effectively rebut one point only to have your opponent wriggle away by saying that your arguments were all fine and stuff, but it was some other point that was really important.
One thing to watch for are unconditional "shoulds" - there are many things that are morally right, or that we should do theoretically, but make sure to point out that things have to looked at in a real-world context, where any given choice always has some harms. Make sure they give you some specifics, so that you can keep things in context. There's a big difference between opposing "The US should try to reduce AIDS in Africa" and "The US should pledge to match contributions to the UN's fund to reduce AIDS in Africa, up to $25 billion". Don't let them trap you in the position of blindly defending something that everyone agrees is a good thing, in principle.
Make Sure It's Fair:
No matter how good a persuasive speaker you are, you're not going to be able to convince another intelligent college student that murder or poverty are good, or that parents should be able to assault their children. If an opponent tries to put you in this position, point out how ridiculous it - a funny, simplified example is usually effective here. Then try to come up with something related that is fair to argue, and suggest it as a more fair debate. Note that you shouldn't do this often - reserve it for those times when things really aren't fair.
Think About the Premises:
Every logical argument is based on a set of assumptions about the world. For example, if I state that "Police entrapment is good because it makes the police more effective in keeping society safe", I am making several critical assumptions: that the police have a right to regulate society, and that societal order is an important goal. I am also (implicitly) making the judgment that safety is more important than individual freedom. One approach would be to challenge the argument directly, and argue that it will not reduce crime. A more effective tactic would be to argue that the policy would adversely affect individual freedom, and then challenge the government's implicit assumption that less crime equals a better society.
Think about the following: on what basic assumptions does the Government rely as the foundations for their case? What is wrong or misguided about those assumptions? How can those assumptions be modified to more closely resemble the situation as you see it? The best arguments look beyond a simple point-by-point rebuttal for deeper flaws or questionable value judgments that can work to your advantage.
Present a Philosophy:
It's a lot easier to make a strong impression if you're arguing for something, rather than merely decrying what is said by your opponent. If you're not able to propose some sort of positive position, you risk looking like you're merely belittling other's ideas, rather than contributing something meaningful. You also make it a lot more difficult for an opponent - they must now not only support their own (presumably well-known) points, but also discredit your philosophy. The opposition philosophy should be a position statement that contradicts the Government case, and it should help create a framework for every argument you're going to make.
Try to think of things that the government hasn't brought up, or has glossed over. The best place to start is usually harms - every policy harms someone, somewhere, and your job is to ferret out just who these will be. Another thing to think of is unintended consequences. Socializing the health care system might increase coverage, but it will also probably increase waiting times for care. You can also make philosophical points - argue why your view of the world is more right than that of your opponent. Weigh the relative impacts of their position and yours. Cite historical examples to support your case. It seems hard at first, but coming up with independent points is something that will develop with practice (followed much, much later by the ability to think of good points).
In debate, it's convention to present three independent points. In reality, at least one or two of these are responses to things that the government said. Try to avoid this (remember, arguing a positive position on opp is a lot stronger place to be in), but it is inevitable at times. Just make sure that you're adding new analysis, not just making fun of your opponent's position. Ideally, your independent points should present a strong enough alternative philosophy that your side emerges as more important, even if all the arguments the government has made are true to some degree.
Make Their Points Sound Stupid:
Unfortunately, that ideal opposition will seldom emerge. The final step is to belittle your opponent's points. Go over everything they've said, and explain why it's either not true, or not as important as they claim. Use examples. Think about if any of their points can be "flipped" to actually support your position. Mostly, just tell the world why they're wrong. This syncs most closely with what most of us do in everyday life, so it's a very difficult skill to teach through anything but practice. The most important think is to remain succinct and keep your speech structured - make sure that people know what your arguments are and what you're responding to through every moment of your speech.