Eco-driving at signalized intersections has significant potential for energy saving. In this paper, we focus on eco-departure operations of connected vehicles equipped with an internal combustion engine and a step-gear automatic transmission. A Bolza-type optimal control problem (OCP) is formulated to minimize engine fuel consumption. Due to the discrete gear ratio, this OCP is a nonlinear mixed-integer problem, which is challenging to handle by most existing optimization methods. The Legendre pseudospectral method combining the knotting technique is employed to convert it into a multistage interconnected nonlinear programming problem, which then solves the optimal engine torque and transmission gear position. The fuel-saving benefit of the optimized eco-departing operation is validated by a passenger car with a five-speed transmission. For real-time implementation, a near-optimal departing strategy is proposed to quickly determine the behavior of the engine and transmission. When a string of vehicles are departing from an intersection, the acceleration of the leading vehicle(s) should be considered to control the following vehicles. This issue is also addressed in this paper.