I am asked to describe the cosets of the subgroup H of $D_n$ (Dihedral group on n vertices) generated by the reflection element s, $H=\langle s \rangle$. So $H=\{id,s\}$, and for $r^i \in D_n ,1 \le i \le n $, $$ r^iH = \{r^i,r^is\}\quad Hr^i =\{r^i,sr^i\}$$ $$sr^iH=\{sr^i,r^{n-i}\} \quad Hsr^i=\{sr^i,r^i\} $$
My question is if this adequately describes the set of cosets of $H \lt D_n$, and if in general when describing cosets, one can separate the cosets into cases as I have done above.
In general there's no nice way of describing all of the cosets for a given group and subgroup. For this specific case, this presentation is just fine.
However, you may worry (as you do in the comments) that you might have missed some of the cosets. Fortunately there is an easy way to check this: all left cosets (and all right cosets) must cover the entire group. So you can make sure there are no more just by guaranteeing that any element of the group is in exactly one left coset and one right coset in your list.
For the dihedral group this is fairly straightforward, since you have a nice simple set of generators and relations.