The idea of Airborne Wind Energy (AWE) is to generate power by flying a tethered airfoil across the windflow. Tethered flight is a fast, strongly nonlinear, unstable and constrained process, motivating control approaches based on fast Nonlinear Model Predictive Control (NMPC) and state estimation approaches based on Moving Horizon Estimation (MHE). In particular, the start-up phase of AWE systems is an involved procedure, and starting and landing using NMPC has not been investigated yet. In this paper, a control strategy for starting-up AWE systems is proposed, based on a rotating carousel that is currently built at the KU Leuven. A computationally efficient 6-DOF control model for a small-scale, rigid airfoil is presented. We present and investigate a control scheme based on receding-horizon Nonlinear Model Predictive Control to track reference trajectories and Moving Horizon Estimation to estimate the actual system state and parameters. The MHE shceme is able to estimate also the wind speed, given no direct wind measurement.