Graph convolutional networks (GCNs) as the emerging neural networks have shown great success in Prognostics and Health Management because they can not only extract node features but can also mine relationship between nodes in the graph data. However, the most existing GCNs-based methods are still limited by graph quality, variable working conditions, and limited data, making them difficult to obtain remarkable performance. Therefore, it is proposed in this paper a two stage importance-aware subgraph convolutional network based on multi-source sensors named I²SGCN to address the above-mentioned limitations. In the real-world scenarios, it is found that the diagnostic performance of the most existing GCNs is commonly bounded by the graph quality because it is hard to get high quality through a single sensor. Therefore, we leveraged multi-source sensors to construct graphs that contain more fault-based information of mechanical equipment. Then, we discovered that unsupervised domain adaptation (UDA) methods only use single stage to achieve cross-domain fault diagnosis and ignore more refined feature extraction, which can make the representations contained in the features inadequate. Hence, it is proposed the two-stage fault diagnosis in the whole framework to achieve UDA. In the first stage, the multiple-instance learning is adopted to obtain the importance factor of each sensor towards preliminary fault diagnosis. In the second stage, it is proposed I²SGCN to achieve refined cross-domain fault diagnosis. Moreover, we observed that deficient and limited data may cause label bias and biased training, leading to reduced generalization capacity of the proposed method. Therefore, we constructed the feature-based graph and importance-based graph to jointly mine more effective relationship and then presented a subgraph learning strategy, which not only enriches sufficient and complementary features but also regularizes the training. Comprehensive experiments conducted on four case studies demonstrate the effectiveness and superiority of the proposed method for cross-domain fault diagnosis, which outperforms the state-of-the art methods.