Recommendation with side information has drawn significant research interest due to its potential to mitigate user feedback sparsity. However, existing models struggle with generalization across diverse domains and types of side information. In particular, three challenges have not been addressed, and they are (1) the diverse formats of side information, including text sequences. (2) The diverse semantics of side information that describes items and users from multi-level in a context different from recommendation systems. (3) The diverse correlations in side information to measure similarity over multiple objects beyond pairwise relations. In this paper, we introduce GENET (Generalized hypErgraph pretraiNing on sidE informaTion), which pre-trains user and item representations on feedback-irrelevant side information and fine-tunes the representations on user feedback data. GENET leverages pre-training as a means to prevent side information from overshadowing critical ID features and feedback signals. It employs a hypergraph framework to accommodate various types of diverse side information. During pre-training, GENET integrates tasks for hyperlink prediction and self-supervised contrast to capture fine-grained semantics at both local and global levels. Additionally, it introduces a unique strategy to enhance pre-training robustness by perturbing positive samples while maintaining high-order relations. Extensive experiments demonstrate that GENET exhibits strong generalization capabilities, outperforming the SOTA method by up to 38% in TOP-N recommendation and Sequential recommendation tasks on various datasets with different side information.