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My question sheet asks about whether a surjective homomorphism exists between various symmetric groups and various $Z_n$ groups, for example between $S_3$ and $Z_3$, or $A_4$ and $Z_3$. To be honest, I don't really know where to start at all - I've been racking my brain for some property of homomorphisms to do with cardinality, or something like that, but I basically have no idea what I'm doing. Any pointers?

Sounds like a fun problem. Basically, your strategy is going to be to write one down, or prove it doesn't exist.

The biggest thing that's useful is knowing how many normal subgroups a group has. For example, $S_n$ has only itself, $1$, and $A_n$. By the first isomorphism theorem, the number of groups that can receive a surjective homomorphism from $S_n$ is severely limited.

A good place to start would be the first isomorphism theorem: if $ f: Grightarrow H$ is a homomorphism, then $G/ker fsimeq operatornameIm f$. So if $f : G rightarrow H$ is a surjective homomorphism, then $G / ker f simeq H$ since surjectivity implies $operatornameIm f = H$. So to determine whether such a homomorphism exists, you should determine whether or not there exists a normal subgroup $G’ leq G$ such that $G/ G’ simeq H$. If such a $G’$ exists, $varphi : G/G’ rightarrow H$ is an isomorphism, and $pi : G rightarrow G/G’$ is the quotient map, then $varphi circ pi$ is a surjective homomorphism from $G$ to $H$.