Group Theory Example What is an example of a group $G$ and a subgroup $H$ such that $|G : H|$ is infinite? I am unsure of how to approach it as it was given as an open ended exercise. Would the group $\mathbb{Z}$ and any subgroup work?

No, most subgroups of $\mathbb{Z}$ are of the form $n\mathbb{Z}$, which have index $n$. There is one subgroup of $\mathbb{Z}$ which has infinite index, the trivial group. It will have infinite index in any infinite group.
For other options, note that the group $\mathbb{Q}$ is not finitely generated, which means that $\mathbb{Z}$ will have infinite index in $\mathbb{Q},$ for example. Or $\mathbb{Z}$ as a subgroup of $\bigoplus^\infty\mathbb{Z}.$ For a finitely generated option, the singly generated group $\mathbb{Z}$ in the doubly generated free product $\mathbb{Z}*\mathbb{Z}$ has infinite index. Or even just $\mathbb{Z}$ as a subgroup of $\mathbb{Z}\oplus\mathbb{Z}$ has infinite index.
