Fast and automated image quality assessment (IQA) for diffusion MR images is crucial so that a rescan decision can be made swiftly during or after the scanning session. However, learning this task is challenging as the number of annotated data is limited and the annotated label is not always perfect. To this end, we introduce an automatic multi-stage IQA method for pediatric diffusion MR images. Our IQA is performed in three sequential stages, i.e., slice-wise quality assessment (QA) using a nonlocal residual network, volume-wise QA by agglomerating the extracted features of slices belonging to one volume using a nonlocal network, and subject-wise QA by ensembling the QA results of volumes belonging to one subject. In addition, we employ semi-supervised learning to make full use of a small amount of annotated data and a large amount of unlabeled data to train our network. Specifically, we first pre-train our network using labeled data, which are iteratively expanded by labeling the unlabeled data with the trained network. Furthermore, we devise a self-training strategy which iteratively relabels and prunes the labeled dataset when training the network to deal with noisy labels. Experimental results demonstrate that our network, trained using only samples of modest size, exhibits great generalizability and is capable of conducting large-scale rapid IQA with near-perfect accuracy.