Is there a systematic way of finding the conjugacy class and/or centralizer of an element? Is there a systematic way of finding the conjugacy class and centralizer of an element? Could the task be simplified if we are working with "special groups" such as $S_n$ or $A_n$? Are there any intuitive approaches? Thanks.

For a finite group, there is a perfectly systematic way: to find the conjugacy class of $x$, just compute every element $g^{-1}xg$, and to find the centralizer, just compare $gx$ to $xg$ for all $g$ in the group. 
For $S_n$, two elements are conjugate if and only if they have the same cycle structure. So $(123)(45)$ is conjugate to $(396)(47)$ in $S_9$, but not to $(12)(45)$ or $(12345)$ or....
There are lots of ways to use other facts you may know about a group or about an element in a group to simplify the calculation of a conjugacy class or a centralizer, but I'm afraid there is no systematic way to list all these facts. They just come with doing examples or reading worked examples in textbooks. 
