The "arguments" presented by religious people generally fall into five categories: Circular reasoning; arguments from authority; implicit admissions that religion is a delusion; lies; and threats. The first step in any discussion is deciding which of these categories best describes the argument being presented. Once you know the basis for the argument, then you know it's weakness because each of these categories is inherently a logical fallacy.
The next step is presenting that weakness in a way that will have an impact on the believer. To do this, you have to make sure it resonates emotionally with the believer and you have to make sure it is short. Mark Twain referred to this idea as "a minimum of sound to a maximum of sense."
Remember two things: Believers are much more aware of the emotional content of what is being said than non-believers usually are, and they will not usually retain anything that takes longer to say than the standard 7 second soundbite. After that time, their short term memory will be full and their attention will start to turn to their reply--even though they haven't really heard what the non-believer is saying. The longer you take to refute a point, the less effective you will be.
Circular reasoning is based on an assumption that the conclusion is true. Without that assumption, the reasoning fails. The argument that existence proves god is a classic case. As is the argument from personal incredulity, in which the believer says he can't believe the beauty and complexity of the universe wasn't designed and created. Both of these arguments are based on the assumption that the existence of a god is the only possible explanation. Thus, the conclusion is implicitly assumed by the argument in order to reach the conclusion.
When faced with a circular argument, simply say that the reasoning is circular and, if necessary, explain why it is. If that doesn't work, then point out that there is no reason to assume that "magic" (which is what the god hypothesis really is) should even be included in the set of possible explanations for the universe.
Arguments from authority can be countered simply by citing examples of where the authorities have been wrong in the past. Every church and every religion has its share of those incidents, which all but the most fanatical believers will admit.
Implicit admissions that religion is a delusion can be met with a simple statement to that effect: Believer: "I just can't live with the idea that my life has no meaning." Atheist: "So, you admit that religion is a delusion?"
Lies have to be countered with truth, even though it probably won't work. If the believer is so fanatical that he or she can't see objective facts, then pointing them out may be a waste of time. Do it anyway, however, because sometimes it can work and even have a profound effect on a believer. If a person starts to recognize that his or her religious leaders are feeding him lies about some things, then he may soon realize that they are not reliable at all.
Threats can be either implicit or explicit. If explicit, take appropriate steps to secure your safety. Such a person is not sane, and it would be better to have absolutely no further contact with him or her. If the threat is implicit such as the threat of damnation, then point out that belief based on fear is extorted and not true belief. If the believer says that god would be satisfied with extorted belief, then point out that he seems to be saying that god is a psychopathic bully who is satisfied by the lies of yes men.
(By the way, I realize that these brief posts leave loose ends and finer points unaddressed. That is the nature of these brief posts, however. I will attempt to tie up the loose ends and address the finer points in later posts. I am, after all, just getting started in this format.)