What does it mean, topologically, to have a simple fundamental group? For instance, the torus $S^1 \times S^1$ has $\mathbb Z \times \mathbb Z$not simple. The case of $S^1$ is $\mathbb Z$, not simple either.
But if instead, in $S^1$ we identify antipodal points, or even, if we identify points after rotating 120 degrees, we would get $\mathbb Z/2\mathbb Z, \mathbb Z/3\mathbb Z$... We can obtain the torus from a $2$ dimensional square by two identifications of a segment, whereas we may construct S$^1$ by doing so from one segment. Is there a relationship between what we identify and being or not simple the fundamental group? Another construction of $S^1$ is by identifying, in $\mathbb R$, integers...May be if we identify elements of some infinite simple group (and still obtain a topological space)...
I'm not sure where you're going with the identification idea. In general there are many ways to construct a given space by making an identification from some other space. Also, the identifications you suggest on $S^1$ do not produce the fundamental groups $\Bbb{Z}/n$; the resulting space is still $S^1$, so the fundamental group is still $\Bbb{Z}$. Maybe it would help you to gather more examples. (You might consider the lens spaces.)
However, there is a correspondence between covering spaces and subgroups of the fundamental group. This gives a topological description of spaces with simple fundamental group: they are spaces which have only two covering spaces, the trivial cover and the universal cover.