Mining natural associations from high-dimensional spatiotemporal signals plays an important role in various fields including biology, climatology, and financial analysis. However, most existing works have mainly studied time-independent signals without considering the correlations of spatiotemporal signals that achieve high learning accuracy. This paper aims to learn graphs that better reflect underlying data relations by leveraging the long- and short-term characteristics of spatiotemporal signals. First, a spatiotemporal signal model is presented that considers both spatial and temporal relations. In particular, we integrate a low-rank representation and a Gaussian Markov process to describe the temporal correlations. Then, the graph learning problem is formulated as a joint low-rank component estimation and graph Laplacian inference. Accordingly, we propose a low rank and spatiotemporal smoothness-based graph learning method (GL-LRSS), which introduces a spatiotemporal smoothness prior into time-vertex signal analysis. By jointly exploiting the low rank of long-time observations and the smoothness of short-time observations, the overall learning performance can be effectively improved. Experiments on both synthetic and real-world datasets demonstrate substantial improvements in the learning accuracy of the proposed method over the state-of-the-art low-rank component estimation and graph learning methods.