The first and most important way to do well on an AP exam is to consistently and deliberately pay attention in class. If you learn things right the first time, you won't be freaking out when May rolls around. Listen to your instructor, participate in class, and most importantly, do your homework as early as possible so that you have plenty of time to figure out problems you don't understand.
You need to decide when to explicitly start studying for the AP exam. If your AP teacher is a good teacher, he/she will have already given you practice AP exams, maybe even as a normal examination in the class. Make sure you give yourself enough time to study so that you can perfect every single concept that will be on the exam.
The best way to study is to start with the general concepts and then work your way down to the specifics. The first time you study for the AP exam, the best thing to do is review your class syllabus / class notes and simply make a list of everything you feel you need to review. Once you've made that list, your studying will be much, much easier.
Over the course of your study time, go through your list and make sure you learn everything on it. You can cross things out once you understand them well; this will show you the progress that you're making on your studying.
Come exam day, you should get a good night's sleep and eat well so that you stay alert, but also keep in mind that if you studied enough, you will know the concepts so well that you won't need to come in with a full 8 hours of sleep. Don't worry if you don't sleep well the night before the exam; again, if you studied, you're going to be fine.
When you actually take the exam, simply read the questions carefully and answer what is asked. In a multi-part problem, if you don't know the answer to a part (and that answer is used in a subsequent part), simply make up the first answer and go from there. The AP graders do not penalize you multiple times for the same mistake; they want to see that you know what you're doing.