I'm considering $\mathbb{Z}_{12}=\{0,1,2,3,4,5,6,7,8,9,10,11\}$, with the operation of addition modulo 12.
To find the distinct subgroups of this, I assume that I would attempt to find each cyclic subgroup $\left<0\right>,\left<1\right>,\left<2\right>,\dots,\left<11\right>$ and find which of these are unique? That is, $\forall a\in\mathbb{Z}_{12},(\left<a\right>=\{\varepsilon, a, a^2, \dots, a^{n-1}\})$
For example, $\left<0\right>=\varepsilon$, $\left<1\right>=\mathbb{Z}_{12}, \left<2\right>=\{0,2,4,6,8,10\}$ and so on.
If I'm not on the right track, how should I do this?
Yes, this approach will work. Something to keep in mind which will make things go faster is that $\langle n\rangle=\langle m\rangle\iff (n,12)=(m,12)$, where $(a,b)$ is the greatest common divisor of $a$ and $b$.