Urban driving with connected and automated vehicles (CAVs) offers potential for energy savings, yet most eco-driving strategies focus solely on longitudinal speed control within a single lane. This neglects the significant impact of lateral decisions, such as lane changes, on overall energy efficiency, especially in environments with traffic signals and heterogeneous traffic flow. To address this gap, we propose a novel energy-aware motion planning framework that jointly optimizes longitudinal speed and lateral lane-change decisions using vehicle-to-infrastructure (V2I) communication. Our approach estimates long-term energy costs using a graph-based approximation and solves short-horizon optimal control problems under traffic constraints. Using a data-driven energy model calibrated to an actual battery electric vehicle, we demonstrate with vehicle-in-the-loop experiments that our method reduces motion energy consumption by up to 24 percent compared to a human driver, highlighting the potential of connectivity-enabled planning for sustainable urban autonomy.