Start by writing all elements of S_4. To be more precise A_4 is a subgroup of S_4 not only a set.
Generally if you want to find elements in S_n then consider all the partations of n, i.e., all the different methods n can be written as sum of positive integers.
Here we have that 4 = 4, 4=3+1, 4=2+2, 4=1+1+2, 4=1+1+1+1. These are the only elements of S_4.
SO in cycle form we write:
Cycle number of elments of this cycle type
It can be little bit difficult for you to find number of elements of a given cycle type because you need some combinatoirs for that, but you will get use to it.
A permutation is even if it can be written as a product of even number of permutations.
I hope this helps, otherwise write again.