General theory of curves

Yeah, that seems to be it. Is there a general theory of classification?

Not a general one, but theres tons of work in that direction. It won't be real curves in R^2 either, it will be complex curves in CP^2 which are much more comfortable and expressive

I watched a handful of infinity this weekend. It was awful. I wouldn't trust the content further than it mentions actual problems/fields.

I see. I guess that makes sense. It would be fun if there was an underlying pattern across all algebraic curves but the fact that just for n=3 there are like 200 different types of cubic curves hints at how completely random this stuff us.

Potentially dumb answer: algebraic geometry?

Yes. Classical algebraic geometry in particular is what OP wants. Try Fulton's Algebraic Curves.

Differential equations are sufficient for describing most curves, and calculus offers some closed form smooth solutions.
Algebraic geometry, which was mentioned before, offers a more precise approach, but is more difficult to analyse.
There are also some topological proofs for curves, but I think they also assume smoothness.

>And he mentioned that Newton found 72 different types of cubic curve (as opposed to the 3 types of quadratics)

That wasn't newton but some sand nigger in the middle east.

There's a number of ways to go about this, mainly algebraic topology, differential geometry and algebraic geometry, depending on what you mean by classify.
If you mean curves defined by polynomial equations, you can look at classical algebraic geometry (Dolgachev's "Classical Algebraic Geometry: A Modern View" is a future classic, you can look there but be warned, it's tough). It has been a staple of mathematics for centuries and is, in some sense, still very active (see the minimal model program for example).

he means algebraic (polynomial) curves. the differentiable viewpoint is very easy: every curve is uniquely determined by its curvature and torsion (up to a euclidean motion)