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Given a graph dataset, how can we generate meaningful graph representations that maximize classification accuracy? Learning representative graph embeddings is important for solving various real-world graph-based tasks. Graph contrastive learning aims to learn representations of graphs by capturing the relationship between the original graph and the augmented graph. However, previous contrastive learning methods neither capture semantic information within graphs nor consider both nodes and graphs while learning graph embeddings. We propose TAG (Two-staged contrAstive curriculum learning for Graphs), a two-staged contrastive learning method for graph classification. TAG learns graph representations in two levels: node-level and graph level, by exploiting six degree-based model-agnostic augmentation algorithms. Experiments show that TAG outperforms both unsupervised and supervised methods in classification accuracy, achieving up to 4.08% points and 4.76% points higher than the second-best unsupervised and supervised methods on average, respectively.