Linear differential equations are a great introduction to eigenvalues. There's a chicken-and-egg problem understanding this, and it would be helpful to take ODEs and Linear Algebra together.
As I first read in Feller, generating functions are a formation that runs the full length of mathematics and beyond; the Laplace transform from ODEs is just a continuous version of a generating function.
As Marcel-Paul Schützenberger first made clear, the notions of "rational" and "algebraic" from pure mathematics match up with finite state machines and push-down automata from computer science. One studies these most easily using generating functions. As Richard Stanley explains in Enumerative Combinatorics Vol. 2, there's a third class of generating functions beyond algebraic: D-Finite. Roughly, this corresponds to how one learns to expand the solution to a differential equation in ODEs as a power series. The true significance of this class in both pure mathematics and computer science is poorly understood, but there's something clearly out there as significant as rational and algebraic. That is intrinsically fascinating.
When I see someone filling a blackboard with equations, I see someone flailing because they don't know what to do. The best math is simple, and can be expressed as philosophy, like recognizing this formation crossing the sky.