Classifying network traffic is the basis for important network applications. Prior research in this area has faced challenges on the availability of representative datasets, and many of the results cannot be readily reproduced. Such a problem is exacerbated by emerging data-driven machine learning based approaches. To address this issue, we present $(Net)^2$ database with three open datasets containing nearly 1.3M labeled flows in total, with a comprehensive list of flow features, for the research community. We focus on broad aspects in network traffic analysis, including both malware detection and application classification. As we continue to grow them, we expect the datasets to serve as a common ground for AI driven, reproducible research on network flow analytics. We release the datasets publicly and also introduce a Multi-Task Hierarchical Learning (MTHL) model to perform all tasks in a single model. Our results show that MTHL is capable of accurately performing multiple tasks with hierarchical labeling with a dramatic reduction in training time.