Diabetic retinopathy (DR) is a leading cause of permanent blindness among the working-age people. Automatic DR grading can help ophthalmologists make timely treatment for patients. However, the existing grading methods are usually trained with high resolution (HR) fundus images, such that the grading performance decreases a lot given low resolution (LR) images, which are common in clinic. In this paper, we mainly focus on DR grading with LR fundus images. According to our analysis on the DR task, we find that: 1) image super-resolution (ISR) can boost the performance of both DR grading and lesion segmentation; 2) the lesion segmentation regions of fundus images are highly consistent with pathological regions for DR grading. Based on our findings, we propose a convolutional neural network (CNN)-based method for joint learning of multi-level tasks for DR grading, called DeepMT-DR, which can simultaneously handle the low-level task of ISR, the mid-level task of lesion segmentation and the high-level task of disease severity classification on LR fundus images. Moreover, a novel task-aware loss is developed to encourage ISR to focus on the pathological regions for its subsequent tasks: lesion segmentation and DR grading. Extensive experimental results show that our DeepMT-DR method significantly outperforms other state-of-the-art methods for DR grading over three datasets. In addition, our method achieves comparable performance in two auxiliary tasks of ISR and lesion segmentation.