I'm learning the theory of commutative modules, but I am not able to comphrehend the generality of it since my stack of examples is too shallow (I only check a theorem against the $\mathbb{R}$-module $\mathbb{R}^2$, $\mathbb{R}^3$ or the $\mathbb{Z}[\sqrt{-5}]$-module $\mathbb{Z}[\sqrt{-5}]$ or the $\mathbb{Z}$-module lattice points in $\mathbb{R}^2$ or $\mathbb{R}^3$).
What are some nice/natural examples of modules over commutative rings with rich structure to keep in mind while learning it?