Visualising a group structure. I've found something really interesting, but it seems to "big" to investigate.
If you take the direct product of some cyclic groups, sometimes interesting things happen.
Let $C_n$ be a cyclic group of order $n$, if you consider $C_n\times C_m$ when $n$ and $m$ are co-prime something interesting happens.
I proved that the result is (isomorphic to a) cyclic group if $m$ and $n$ are co-prime already, this is about something more interesting.
I use $\langle g\rangle$ to denote the group $\{g^0,g^1,g^2,...,g^k\}$
A pattern emerges when you consider things like $\langle(g_1,g_2)\rangle$. I used the multiplicative modulo groups (as I know (if $n$ is prime at least) that these can be generated/are cyclic -can someone confirm this for all?) 
The group $C_2 \times C_2$ is interesting, it gives:
$(1,1)$
$(1,2)\rightarrow(1,1)\rightarrow(1,2)$
$(2,1)\rightarrow(1,1)\rightarrow(2,1)$
$(2,2)\rightarrow(1,1)\rightarrow(2,2)$
Which can be drawn nicely as a graph ($(1,1)$ at the centre, surrounded by the nodes $(1,2)$ $(2,1)$ and $(2,2)$ with a line coming from $(1,1)$ to it, and another arc going back)
I'm curious as to the larger pattern here, but if I try ... say $C_4 \times C_4$ (from multiplication under modulo $5$ for instance) what do I get? There are $16$ different "chains" to generate, this would take a while.
Is there a better way to explore this? I am really curious but right now Graphviz and a python script seem like a good way, but this just shows the structure, it doesn't really explore it! 
It is also interesting that the graph is planar, well I think so anyway. What have I discovered?
 A: Yes, it's big, leading to whole "crossover" subjects like geometric group theory or algebraic graph theory.
Here is an interesting result related to your topic of Cayley graphs of finite abelian groups.  The spectrum of a Cayley graph $(G, S)$ of a finite abelian group $G$ is integral if and only if the generator set $S$ is a union of subgroup cosets.  
This book (for "beginners") shows the usefulness of Cayley graphs (and their spectra) in computer science and engineering as expander graphs.
For planar Cayley graphs of finite groups, see the classical reference mentioned, for example, here.  A proof can be found here (p. 287).

The only [finite] groups that can give planar Cayley graphs are exactly $Z_n$, $Z_2×Z_n$, $D_n$, $S_4$, $A_4$, and $A_5$, as proved by Maschke (1896).

Googling "planar Cayley graph" returns very recent research results about infinite groups, enumerating and characterizing infinite planar Cayley graphs in terms of group presentations, and the implications of Cayley graph connectivity to planarity.
