Graph neural networks (GNNs) have shown great power in learning on graphs. However, it is still a challenge for GNNs to model information faraway from the source node. The ability to preserve global information can enhance graph representation and hence improve classification precision. In the paper, we propose a new learning framework named G-GNN (Global information for GNN) to address the challenge. First, the global structure and global attribute features of each node are obtained via unsupervised pre-training, and those global features preserve the global information associated with the node. Then, using the pre-trained global features and the raw attributes of the graph, a set of parallel kernel GNNs is used to learn different aspects from these heterogeneous features. Any general GNN can be used as a kernal and easily obtain the ability of preserving global information, without having to alter their own algorithms. Extensive experiments have shown that state-of-the-art models, e.g., GCN, GAT, Graphsage and APPNP, can achieve improvement with G-GNN on three standard evaluation datasets. Specially, we establish new benchmark precision records on Cora (84.31%) and Pubmed (80.95%) when learning on attributed graphs.