The increasing penetration of renewable energy introduces variability and uncertainties in power systems. These challenges can be mitigated through the coordinated operation of gas and electricity resources, leveraging the flexibility of gas-fired power plants and the inherent storage capacity of gas pipelines. A key difficulty of this approach lies in accurately modeling gas flow dynamics that are typically represented by the non-convex Weymouth equation. To address this, the paper presents a two-stage Learning-to-Optimize (L2O) framework that bypasses the complexity of explicit gas flow modeling, enabling fast and reliable solutions for the coordinated day-ahead power and gas dispatch problem. The proposed L2O process uses advanced recurrent neural network architectures to forecast the next-day operational states of the gas system based on historical system data. Then, the predicted operating points are used to linearize the Weymouth equation, allowing the resulting linear constraints to be embedded into the optimization problem for improved tractability. We finally introduce a trust region to constrain solutions within a physically valid neighborhood of the predicted state. Numerical results demonstrate that our L2O approach achieves cost efficiency comparable to state-of-the-art formulations, while significantly reducing computation time.