Mark the places where the unit circle intercepts the X and Y axes. The unit circle is the circle with radius one, centered at the origin. Rotation starts at the X axis and goes counterclockwise. The X intercept should be marked with a 0 at point (1, 0). Rotating one quarter of the way around the circle -- up to the positive Y axis -- label the point (0, 1) with pi/2 as this is 1/4 of 2 pi. Continuing on until the negative X axis is reached, label (-1, 0) pi because it is halfway around the circle and 1/2 of 2 pi is pi. Similarly, the point (0, -1) on the negative Y axis is labeled 3pi/2.
Memorize the markings between 0 and pi/2 -- these will be the basis of all the other markings. The first of these marks is half way between 0 and pi/2 -- the 45-degree angle. This mark is 1/2 X pi/2 = pi/4. The other two marks are the 1/3 and 2/3 marks which correspond to 30 degrees and 60 degrees. these are 1/3 X pi/2 = pi/6 and 2/3 X pi/2 = pi/3. To summarize: The markings in the first quadrant -- between 0 and pi/2 -- are pi/6, pi/4 and pi/3. These should be thoroughly understood and completely memorized before trying to understand the markings in the other quadrants.
Learn the markings in the other quadrants by adding the marks in the first quadrant to those at the intercept that begins the quadrant you are interested in. For example, if you are learning the markings in the third quadrant -- where all the points have two negative coordinates -- start with the mark that begins the third quadrant; the pi mark at (-1,0). Now add the three marks in the first quadrant to get the three marks in the third quadrant, pi + pi/6 = 7pi/6, pi + pi/4 = 5pi/4 and pi + pi/3 = 4pi/3. All of these calculations are easy enough that you can do them mentally if you are familiar with the markings in the first quadrant.