"Natural" example of cosets

The plane $\mathbb{R}^2$ is a group under addition, and the $x$-axis $\{(a,0)\colon a\in\mathbb{R}\}$ is a subgroup of it. Then the lines parallel to $x$-axis are precisely the cosets of this subgroup.

Instead of $x$-axis, you can take any line through origin; it will be a subgroup, and lines parallel to it will be cosets.

Similarly, $\mathbb{C}^*=\mathbb{C}-\{0\}$ is group under multiplication; think it like a punctured plane. Then $S^1=\{z\in\mathbb{C}\colon |z|=1\}$ is a subgroup, which is a circle with center origin and radius $1$. Its cosets are concentric circles to $S^1$.

Edit: Consider the group ${\rm GL}_n(k)$ of $n\times n$ invertible matrices over a field $k$ and ${\rm SL}_n(k)$ be the subgroup consisting of matrices with determinant $1$. Then for every $\lambda\in k-\{0\}$, the subset of ${\rm GL}_n(k)$ consisting of matrices with determinant $\lambda$ is a coset of ${\rm SL}_n(k)$ (where $\lambda=1$ gives trivial coset).

Probably the example most students will find the most familiar is the set of cosets of the integers modulo some fixed integer.

So for an integer $n$, the cosets of the subgroup $n\mathbb{Z}$ in $\mathbb{Z}$ consists of subset of the form $[a] = \{x\in \mathbb{Z}\mid x\equiv a \pmod n\}$ and if we pick one for each $a$ with $0\leq a\leq n-1$ then we get all the cosets.