In this paper, we deal with output tracking control problems for input-affine nonlinear systems. We propose a deep learning-based solution whose foundations lay in control theory. We design a two-step controller: a contraction-based feedback stabilizer and feedforward action. The first component guarantees convergence to the steady-state trajectory on which the tracking error is zero. The second one is inherited from output regulation theory and provides forward invariantness of such a trajectory along the solutions of the system. To alleviate the need for heavy analytical computations or online optimization, we rely on deep neural networks and link their approximation error to the tracking one. Mimicking the analytical control structure, we split the learning task into two separate modules. For the stabilizer module, we propose a switching objective function balancing feasibility of the solution and performance improvement. We test our solution in a challenging environment to validate the proposed design.