Characterizing the interactions and dynamics of quantum mechanical systems is an essential task in the development of quantum technologies. We propose a novel protocol for estimating the underlying Hamiltonian dynamics and Markovian noise of a multi-qubit device. It is based on the efficient estimation of the time-derivatives of few qubit observables using polynomial interpolation. For finite range dynamics, our protocol exponentially improves the necessary time-resolution of the measurements and quadratically reduces the overall sample complexity compared to previous approaches. Furthermore, we show that our protocol can characterize the dynamics of systems with algebraically decaying interactions. The implementation of the protocol requires only the preparation of product states and single-qubit measurements, rendering it applicable for the characterization of both current and future quantum devices.