Huge computational costs brought by convolution and batch normalization (BN) have caused great challenges for the online training and corresponding applications of deep neural networks (DNNs), especially in resource-limited devices. Existing works only focus on the convolution or BN acceleration and no solution can alleviate both problems with satisfactory performance. Online training has gradually become a trend in resource-limited devices like mobile phones while there is still no complete technical scheme with acceptable model performance, processing speed, and computational cost. In this research, an efficient online-training quantization framework termed EOQ is proposed by combining Fixup initialization and a novel quantization scheme for DNN model compression and acceleration. Based on the proposed framework, we have successfully realized full 8-bit integer network training and removed BN in large-scale DNNs. Especially, weight updates are quantized to 8-bit integers for the first time. Theoretical analyses of EOQ utilizing Fixup initialization for removing BN have been further given using a novel Block Dynamical Isometry theory with weaker assumptions. Benefiting from rational quantization strategies and the absence of BN, the full 8-bit networks based on EOQ can achieve state-of-the-art accuracy and immense advantages in computational cost and processing speed. What is more, the design of deep learning chips can be profoundly simplified for the absence of unfriendly square root operations in BN. Beyond this, EOQ has been evidenced to be more advantageous in small-batch online training with fewer batch samples. In summary, the EOQ framework is specially designed for reducing the high cost of convolution and BN in network training, demonstrating a broad application prospect of online training in resource-limited devices.