Multiple-choice exams are popular amongst teachers (at least in Spain), because they are easy to grade and there's no discussion to them. But it should be noted that it's terribly dificult to write a fair exam, and most of the time one can get a better grade than deserved.
First of all, you should analyze the grading system. I have seen many tests carry a flawed score. One would expect some numbers that yield a 0 average if you answer at random. For example, for a 4-choice question, an incorrect answer should substract one third on the score for a correct answer, so that a random choice would make:
being 'a' the score for a correct answer. That's the average score you'd get if you answered each option with equal probability. Anything less than that, makes you consider answering questions randomly (I have even seen exams where wrong answers did not substract, you should answer all the questions in that case). Oh! Some exams have questions with a variable number of options, check that.
Stretching a bit further than that, you should consider the possibilities of answering randomly if you've narrowed down to some choices. Let's say that we've got five options, we get one point for the correct answer and -0.25 (it's fair) for an error. If we know that only two options are possible, we'll score 0.5*1+0.5*(-0.25)=0.375 on the average. That's positive! In fact, on a fair exam, eliminating just one option will grant a positive average.
The problem with that is that it adds randomness to your score. If you know you are going to fail, it can be a risk worth taking; or if you know you're going to pass with some margin, you can risk it to get a higher grade, but that could backfire.
Another thing to consider is the number of questions in the exam. As the number of questions in the exam grows, the chances to have 'good luck' or 'bad luck' decrease quickly. If you are answering some questions randomly (as in our previous example) so that you get a positive average score in them, you can expect that to work better as the number of question grows.
Also, if you're working for a given score (says, 5 out of 10 to pass), take into account the number of questions you have answered. Suppose we've got 20 questions and you need 10 questions right to pass, and wrong answers score -0.25. If you answer 12 questions, you can only answer one wrong (that's 10.75, and 9.5 if you get two wrong). If you answer 13, you can get two wrong (for 10.5), so if you get one extra right randomly, it would compensate another error for free!.
Regarding the options given, you always should apply whatever logic you have at hand. Look out carefully for the wording, because it is very common amongst some teachers to play a lot with words to make some questions difficult. And it is very easy to write some options wrong and easily eliminated.
Analyze the options in pairs too, sometimes you'll find that two options are totally complementary, that is, that one of them must be correct (if an option says 'A' is 'B' and another says 'A' is not 'B', for example), and you can easily discard the other options.
Meta-options are also to be considered (options like 'none of the above are correct', '(a) and (b) are correct', etc.). For starters, sometimes they are incorrectly written and contradictory, and you can play on that. The other thing to consider is the dreaded 'none of the above'. You should realize that that option is very harmful. None of the above nearly never can be discarded by plain logic (you must work out that one of the other options must be correct, like in the 'complementary' above case), so it limits the narrowing to two choices by default. If possible, try to get copies of old exams from that teacher (that is terribly useful always) and analyze his/her policy on 'none of the above'; most have strong tendencies on them that you can play to your advantage (but that is very risky).
Well, maybe that's all my two cents. I guess studying also helps. Also, I'd appreciate comments, especially about other kinds of multiple-choice I have no experience with (exams where there is more than one correct option, etc.).