How can we accurately classify graphs? Graph classification is a pivotal task in data mining with applications in social network analysis, web analysis, drug discovery, molecular property prediction, etc. Graph neural networks have achieved the state-of-the-art performance in graph classification, but they consistently struggle with overfitting. To mitigate overfitting, researchers have introduced various representation learning methods utilizing graph augmentation. However, existing methods rely on simplistic use of graph augmentation, which loses augmentation-induced differences and limits the expressiveness of representations. In this paper, we propose AugWard (Augmentation-Aware Training with Graph Distance and Consistency Regularization), a novel graph representation learning framework that carefully considers the diversity introduced by graph augmentation. AugWard applies augmentation-aware training to predict the graph distance between the augmented graph and its original one, aligning the representation difference directly with graph distance at both feature and structure levels. Furthermore, AugWard employs consistency regularization to encourage the classifier to handle richer representations. Experimental results show that AugWard gives the state-of-the-art performance in supervised, semi-supervised graph classification, and transfer learning.