To find conjugacy classes, simply take every element and conjugate it by every other element! This will tell you which elements are conjugate!

However - there are some `tricks' to save you time. The main one is that conjugacy classes are disjoint, so is g is conjugate to h then g and h are in the same conjugacy class (obviously). So, take an element and work out all its conjugates, by conjugating by every other element. The elements it is conjugate to form its conjugacy class. Do the same to an element not in this conjugacy class, and so you will have two conjugacy classes. Do the same to an element in neither of these two conjugacy classes, and so on...

Also, don't forget the identity! It always sits in a conjugacy class of its own.