I would say that broadly there are three main geometric themes in the classical theory of complex analysis in the plane - here we do not consider Riemann surfaces which are better understood in terms of complex manifolds, 2 or more complex variables or topology per se, just what one would consider fairly basic euclidean and hyperbolic geometry.
We have euclidean stuff that is represented in geometric property of domains like convexity, star like and more intricate stuff (spiral like, convex in one direction etc) and then the theory deals with the strong conditions that are imposed on Taylor series with images domains as above (convex etc) - first we prove results for injective (univalent) holomorphic functions and then we use subordination and quasi-subordination theory to get results in general. Herglotz theorem about holomorphic functions on the unit disc with positive real part (or equivalently positive harmonic functions on the disc) is the typical example one learns first - this usually is called geometric function theory
We have hyperbolic geometric stuff where holomorphic functions from the unit disc to the unit disc are contractions in the hyperbolic metric (this is a fancy way of stating Schwarz lemma) - since the hyperbolic metric is pointwise proportional to the Euclidean metric, angles are same under both metrics so conformality (angle preserving) of holomorphic functions at non critical points belongs to both topics
We have boundary properties of holomorphic functions which belong to both topological (shape) and geometric conditions (areas, lengths..) and usually the first results one learns are topological (eg bounded holomorphic functions on the unit disc have non zero radial boundary limits ae) and then the deeper results are geometric
