Hyperspectral image (HSI) denoising is a crucial preprocessing task to improve the performance of the subsequent HSI interpretation and applications. With recent progresses in deep learning, HSI denoising methods based on deep neural networks have attracted increasing interests and achieved the state-of-the-art performance. Nevertheless, most of these methods are based on network structures originally developed for grayscale and color images, and require change of network structure to be applicable to HSIs. The new network architectures often lead to complicated models and limited flexibility, which in turn result in difficulty in learning and demand of a large number of training samples. In this paper, we propose an innovative two-stage learning method including pre-training and fine-tuning procedures. In the first stage, a denoising convolutional neural network can be pre-trained with pairs of corrupted and clean images. In the second stage, the pre-trained network is fine-tuned via a self-supervised learning strategy to capture the spectral correlation in HSIs. The training pairs in the second stage are constructed from the neighboring band images in the target noisy HSI, leading to a novel idea of embedding spectral information into denoiser through target image rather than change of network architecture. This model has strong adaptability such that many image denoising networks can be easily adopted for HSIs, while the external hyperspectral training set is optional but not mandatory. Experimental results show that our method has competitive performance compared with the state-of-the-art approaches.