The best way to prepare for any test is to locate old versions of the exam and practice with those, especially studying the problems you answered incorrectly. This is true for the AMCs and MathCounts. AoPS also offers online prep classes. For those of you who prefer learning from textbooks, here are few recommendations. These books take problems from old competitions and organize them by topic so you can ramp up your skills.
Competition Math for Middle School by J. Batterson. In particular I enjoyed the chapters on counting and probability for their concise and clear introductions to the subjects. You’ll be up to speed on these topics in no time. Other chapters are algebra, geometry, and number theory.
First Steps for Math Olympians by J. Douglas Faires. This is the next step in difficulty is preparing for the AMC 10/12. This book uses questions from those exams as exercises. Content includes: ratios, polynomials, functions, triangles, circles, polygons, counting, probability, primes, number theory, sequences and series, statistics, trig, 3D geometry, logs, and complex numbers.
A Gentle Introduction to the American Invitational Mathematics Exam by Scott Annin. This title seemed to be speaking to me. I qualified for AIME my senior year of high school, but bombed the test, so I’ve been living in fear of the AIME every since. I’m working through the chapter 1 and I’m pleased to find myself actually solving some problems without assistance. Contents include: algebra, combinatorics, probability, number theory, sequences/series, logs, trig, complex numbers, polynomials, plane geometry, and 3D geometry. Nearly all exercises are former AIME problems.