Recently, brain-inspired spiking neural networks (SNNs) have demonstrated promising capabilities in solving pattern recognition tasks. However, these SNNs are grounded on homogeneous neurons that utilize a uniform neural coding for information representation. Given that each neural coding scheme possesses its own merits and drawbacks, these SNNs encounter challenges in achieving optimal performance such as accuracy, response time, efficiency, and robustness, all of which are crucial for practical applications. In this study, we argue that SNN architectures should be holistically designed to incorporate heterogeneous coding schemes. As an initial exploration in this direction, we propose a hybrid neural coding and learning framework, which encompasses a neural coding zoo with diverse neural coding schemes discovered in neuroscience. Additionally, it incorporates a flexible neural coding assignment strategy to accommodate task-specific requirements, along with novel layer-wise learning methods to effectively implement hybrid coding SNNs. We demonstrate the superiority of the proposed framework on image classification and sound localization tasks. Specifically, the proposed hybrid coding SNNs achieve comparable accuracy to state-of-the-art SNNs, while exhibiting significantly reduced inference latency and energy consumption, as well as high noise robustness. This study yields valuable insights into hybrid neural coding designs, paving the way for developing high-performance neuromorphic systems.