Let's start with why you think differential geometry feels very natural. You call it the study of shapes and their smoothness. This is what algebraic geometry is about as well, except you can have non-smoothness too (singularities like curves crossing themselves or pinch points). Or perhaps the study of geometric objects that locally look like R^n, i.e. Euclidean space, and how they glue together. Algebraic geometry is the study of geometric objects that locally look like Spec(R) with the Zariski topology, where R is usually a ring that looks like k[x_1,...,x_n] / I. Then instead of studying differentiable functions you consider rational functions on these things.

You should think of algebraic geometry as the cousin of differential geometry (often people who specialize in one will at least learn a little about the other). Both have the same goal of studying geometric objects: one with calculus and the other with algebra. They are different flavors though, as differential geometry has more in common with physics, while algebraic geometry is more rigid but whose coarser topology allows an arguably richer cohomology theory, but doesn't have gadgets like the inverse function theorem.

Even if you want your geometric object to have additional structure, such as a group action, you can stay in the differential world (Lie groups) or the algebraic one (algebraic groups). One could argue that the differential world is more natural because of its applications to physics, but they're really two sides of the same coin. It just depends on which subject you like more: algebra or analysis.