Graphs provide a powerful framework for representing complex relationships in diverse domains, including social networks, biological systems, transportation, and communication networks. Graph representation learning has emerged as a key technique for extracting meaningful embeddings, enabling tasks such as node classification, graph classification, and community detection. While Graph Neural Networks (GNNs) have significantly advanced this field, many existing methods fail to jointly capture local structures and higher-order dependencies, limiting their effectiveness. To address these challenges, this dissertation proposes novel graph representation learning models that integrate both local and global structural information for improved performance across static and dynamic graphs. In our first work, we introduce a proximity-based graph compression method for network embedding. Our approach leverages neighborhood similarity to compress the input graph into a smaller, more structured representation, preserving local proximity within super-nodes. Learning embeddings on the compressed graph reduces computational costs while maintaining global structural information. We then refine these embeddings back to the original graph, enhancing representation quality. In our Second work, we propose a multi-channel Motif-based Graph Pooling method named (MPool) that captures the higher-order graph structure with motif and also considers the local and global graph structure through a combination of selection and clustering-based pooling operations. In our third work, we propose a graph contrastive learning model with graph compression for graph classification. Using K-core and K-truss-based compression, we generate two views that capture both low-order and higher-order structures. These representations are optimized via contrastive loss and integrated for the final graph classification. In our fourth work, we propose a Dynamic Graph Contrastive Learning (DyGCL) method for event prediction. Our model DyGCL employs a local view encoder to effectively capture the local dynamic structure and a global view encoder to capture the higher-order structure of the dynamic graphs. Our extensive experiments demonstrate that our proposed methods outperform the state-of-the-art methods for different graph mining tasks like node classification, graph classification, and event prediction on various real-world datasets. These contributions advance graph representation learning by effectively integrating local and global structural information, opening new possibilities for more interpretable and scalable models in future research.