With the advancement of electromagnetic interference and counter-interference technology, complex and unpredictable interference signals greatly reduce radar detection, tracking, and recognition performance. In multi-interference environments, the overlap of interference cross-correlation peaks can mask target signals, weakening radar interference suppression capability. To address this, we propose a joint waveform and filter design method called Multi-Interference Suppression Network (MISNet) for effective interference suppression. First, we develop a design criterion based on suppression coefficients for different interferences, minimizing both cross-correlation energy and interference peak models. Then, for the non-smooth, non-convex optimization problem, we use complex neural networks and gating mechanisms, transforming it into a differentiable problem via end-to-end training to optimize the transmit waveform and receive filter efficiently. Simulation results show that compared to traditional algorithms, MISNet effectively reduces interference cross-correlation peaks and autocorrelation sidelobes in single interference environments; it demonstrates excellent robustness in multi-interference environments, significantly outperforming CNN, PSO, and ANN comparison methods, effectively improving radar interference suppression performance in complex multi-interference scenarios.