In this paper, the axial–torsional dynamics and complex nonlinear phenomena, particularly stick–slip vibrations, inherent in drill string systems were investigated. A comprehensive dynamic model incorporating significant nonlinearities, including dry friction and time-delay effects, was developed and discretized using finite element methods. Dynamic behavior under varying operational parameters was simulated, revealing stable, periodic, and chaotic motion can be found in this system. A data-driven approach utilizing deep learning was adopted for motion prediction. Specifically, a 1DCNN-LSTM architecture was developed and successfully employed for the spatial prediction of downhole dynamic at multiple locations along the drill string, based on only the measurable data from wellhead. Furthermore, the temporal prediction of future states of the system was conducted using an LSTM network. For periodic motions, accurate long-term prediction, including the characterization of stick–slip phases, was demonstrated. In the case of chaotic motions, short-term predictions, ranging from 15 to over 25 s depending on system complexity, were achieved. While long-term predictions for chaotic motion are inherently limited, the long-term predicted data were shown to preserve the essential statistical characteristics of the system dynamics, as verified by phase portraits and frequency spectrum analysis. The potential of deep learning methodologies for real-time downhole monitoring based on surface measurements is highlighted by this study.