Nonlinear spectroscopy employs a series of laser pulses to interrogate dynamics in large interacting many-body systems, and it has become a highly successful method for experiments in chemical physics. Current quantum optical experiments approach system sizes and levels of complexity that require the development of efficient techniques to assess spectral and dynamical features with scalable experimental overhead. However, established methods from optical spectroscopy of macroscopic ensembles cannot be applied straightforwardly to few-atom systems. Based on the ideas proposed in M. Gessner, (arXiv:1312.3365), we develop a diagrammatic approach to construct nonlinear measurement protocols for controlled quantum systems, and we discuss experimental implementations with trapped ion technology in detail. These methods, in combination with distinct features of ultracold-matter systems, allow us to monitor and analyze excitation dynamics in both the electronic and vibrational degrees of freedom. They are independent of system size, and they can therefore reliably probe systems in which, e.g., quantum state tomography becomes prohibitively expensive. We propose signals that can probe steady-state currents, detect the influence of anharmonicities on phonon transport, and identify signatures of chaotic dynamics near a quantum phase transition in an Ising-type spin chain. © 2014 American Physical Society.