In this paper, we formulate a full Bayesian approach for spatio-temporal Gaussian process regression under practical conditions such as measurement noise and unknown hyperparmeters (particularly, the bandwidths). Thus, multi-factorial effects of observations, measurement noise and prior distributions of hyperparameters are all correctly incorporated in the computed predictive distribution. Using discrete prior probabilities and compactly supported kernels, we provide a way to design sequential Bayesian prediction algorithms that can be computed (without using the Gibbs sampler) in constant time as the number of observations increases. Both centralized and distributed sequential Bayesian prediction algorithms have been proposed for mobile sensor networks. An adaptive sampling strategy for mobile sensors, using the maximum a posteriori (MAP) estimation, has been proposed to minimize the prediction error variances. Simulation results illustrate the effectiveness of the proposed algorithms.