Skim over the entire exam prior to beginning work.
Don't necessarily do the problems in order. Instead, get those problems out of the way you feel confident you can do quickly and well. Observe how the problems are weighted, and direct your efforts to where you believe you can pick up points most easily. This does not necessarily mean attempting the most heavily weighted problem first; rather, it means first doing the problem for which you can accumulate points at the fastest rate. Indeed, there is a good chance that this is not the most heavily weighted problem, since many instructors dislike giving any one problem significantly greater or fewer points than the average, thereby underweighting the harder problems and overweighting the easier ones.
Before writing on any given problem, think. A small investment in time at the beginning can save time overall (for you might thereby choose a more efficient method of solving the problem).
Do precisely what is requested. In particular, don't waste time doing things that will not receive credit. For example, unless explicitly required, do not rewrite the exam problems on your paper.
Pace yourself through the exam. Example: On a 50-minute exam worth a 100 points, you should be accumulating 2 points per minute; thus, a 26-point problem should be completed in 13 minutes. Do this calculation at the start of the exam if the problem weights are given.
If only for psychological reasons, most graders use nonlinear grading by which the early points of a problem are easier to get:
Therefore, always write something (meaningful) down for every problem, if only a little. At the other end, even with linear grading, there are diminishing returns in terms of points-per-effort in trying to squeeze every last point out of a given problem; if time is low, it may be better to move on.
Communicate with the grader. In particular, if you are running out of time, state the steps you would perform if you were to continue the problem.
Show your work and make clear your reasoning in order to have a chance to receive partial credit.
As with homework, and even more importantly, neatness counts.
In courses on subjective material (e.g., humanities), just regurgitate the material from class and the text(s). Supplying you own opinions may sound good in theory, but it has the risk of running counter to the opinions of the instructor or grader. Conversely, restatements of the class/text material are easy for the grader to recognize as something deserving credit. Remember: Unless the exam is multiple-choice, then a human being---who typically wants to grade the many exams in front of him/her as quickly and painlessly as possible---is doing the grading.
Always check over your answers if you have time.
Further Suggestions
Unify and simplify your knowledge: A textbook presents the subject in a particular form, as does an instructor. By their very natures, however, textbooks and lectures tend to present subjects sequentially. Take the extra step of understanding the material in your terms, which may involve recognizing relationships that could not be conveniently expressed in the order presented in the text(s) and lectures.
Remember, almost every logically consistent topic is simple at its foundation. Try to recognize the simple underlying relationships in the subject at hand; these are often left unstated by instructors and textbooks.
Try to learn general principles and methods. Learning by examples (putting the new in terms of the familiar) can only take you so far.
Learn as many methods of problem-solving as you can. This is especially helpful for exams, when time is of the essence.
Ask yourself questions. Why didn't the instructor or text(s) do this or that? Explore your own ideas. Try to understand the course material in detail.
It is often said that the best way to learn something is to teach it. Do you know the subject matter well enough to explain it clearly and completely to someone else?
Learn by observing others. Notice what works for them and consider incorporating those methods into yourself. Ask yourself "Why didn't I think of that?", and try to develop the related ability.
Attempt to be methodical, neat, legible, deliberate, precise, knowledgeable, and reliable on the one hand, and creative, spontaneous, imaginative, smart, clever, articulate, and flexible on the other. The first mentality thrives on order, and inherently tries to do well what it already knows how to do; the second mentality thrives on disorder, and inherently tries to expand upon its abilities. Adopt the best of these two mentalities. Remember, every tool is a potential crutch. The first mentality may rely too heavily on already-mastered skills; but, the second mentality may fail to carefully apply those same skills.
Think about and question everything, even the statements appearing here (and, yourself!). But, realize that it is equally foolish to be different merely for the sake of being different, as it is to mindlessly conform to the norm.
For maximum efficiency, have several projects going at once. Then, if you get tired, frustrated, or bored working on one item, you can easily move onto something else, thereby staying productive as well as giving pending problems a chance to work themselves out subconsciously.
Anticipate. For example, you may need to ask the instructor about the present assignment, but he/she is only guaranteed to be available at certain times; therefore, you should look over the assignment early.
Forget pulling "all-nighters". These merely amount to borrowing from tomorrow, at which time you will find yourself considerably less functional. All-nighters are really an indication of not having properly planned your activities.
If possible, bring your textbook(s) to class.
Take your lecture notes in pencil, since any modifications can then be made quickly and neatly.
Closing
Overall, there is one basic trait that distinguishes successful students from those that are not:
Successful students force themselves to understand.
They do not merely go through the motions of attending class, reading the text(s), and doing the homework, expecting these actions to necessarily suffice. Rather, they are continually asking, "Do I really understand what's going on here?" They ask this question of themselves honestly, applying an internal barometer formed from experience to detect the slightest lack of understanding, be it ignorance or confusion. And, if the answer is "No", then the situation is viewed as unacceptable, and more effort is the response.
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