This paper presents a novel class of self-organizing sensing agents that learn an anisotropic, spatio-temporal Gaussian process using noisy measurements and move in order to improve the quality of the estimated covariance function. This approach is based on a class of anisotropic covariance functions of Gaussian processes developed to model a broad range of anisotropic, spatio-temporal physical phenomena. The covariance function is assumed to be unknown a priori. Hence, it is estimated by the maximum likelihood (ML) estimator. The prediction of the field of interest is then obtained based on a non-parametric approach. An optimal navigation strategy is proposed to minimize the Cramér-Rao lower bound (CRLB) of the estimation error covariance matrix. Simulation results demonstrate the effectiveness of the proposed scheme.