We extend metric learning by studying the Riemannian manifold structure of the underlying data space induced by similarity measures between data points. The key quantity of interest here is the Riemannian metric, which characterizes the Riemannian geometry and defines straight lines and derivatives on the manifold. Being able to estimate the Riemannian metric allows us to gain insights into the underlying manifold and compute geometric features such as the geodesic curves. We model the observed similarity measures as noisy responses generated from a function of the intrinsic geodesic distance between data points. A new local regression approach is proposed to learn the Riemannian metric tensor and its derivatives based on a Taylor expansion for the squared geodesic distances, accommodating different types of data such as continuous, binary, or comparative responses. We develop theoretical foundation for our method by deriving the rates of convergence for the asymptotic bias and variance of the estimated metric tensor. The proposed method is shown to be versatile in simulation studies and real data applications involving taxi trip time in New York City and MNIST digits.