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Distributed renewable generation, elastic loads, and purposeful manipulation of meter readings challenge the monitoring and control of today's power systems (PS). In this context, to maintain a comprehensive view of the system in real time, fast and robust state estimation (SE) methods are urgently needed. Conventional PSSE solvers typically entail minimizing a nonlinear and nonconvex least-squares by e.g., the workhorse Gauss-Newton method. Those iterative solvers however, are sensitive to initialization and may get stuck in local minima. To overcome these hurdles and inspired by recent image denoising techniques, this paper advocates a learnable regularization term for PSSE that uses a deep neural network (DNN) prior. For the resultant regularized PSSE problem, a "Gauss-Newton-like" alternating minimization solver is first developed. To accommodate real-time monitoring, a novel end-to-end DNN is constructed by unrolling the proposed alternating minimization solver. Interestingly, the power network topology can be easily incorporated into the DNN by designing a graph neural network (GNN) based prior. To further endow the physics-based DNN with robustness against bad data, an adversarial DNN training method is discussed. Numerical tests using real load data on the IEEE $118$-bus benchmark system showcase the improved estimation and robustness performance of the proposed scheme compared with several state-of-the-art alternatives.