Speech Emotion Recognition (SER) has become a growing focus of research in human-computer interaction. An essential challenge in SER is to extract common attributes from different speakers or languages, especially when a specific source corpus has to be trained to recognize the unknown data coming from another speech corpus. To address this challenge, a Capsule Network (CapsNet) and Transfer Learning based Mixed Task Net (CTL-MTNet) are proposed to deal with both the single-corpus and cross-corpus SER tasks simultaneously in this paper. For the single-corpus task, the combination of Convolution-Pooling and Attention CapsNet module (CPAC) is designed by embedding the self-attention mechanism to the CapsNet, guiding the module to focus on the important features that can be fed into different capsules. The extracted high-level features by CPAC provide sufficient discriminative ability. Furthermore, to handle the cross-corpus task, CTL-MTNet employs a Corpus Adaptation Adversarial Module (CAAM) by combining CPAC with Margin Disparity Discrepancy (MDD), which can learn the domain-invariant emotion representations through extracting the strong emotion commonness. Experiments including ablation studies and visualizations on both single- and cross-corpus tasks using four well-known SER datasets in different languages are conducted for performance evaluation and comparison. The results indicate that in both tasks the CTL-MTNet showed better performance in all cases compared to a number of state-of-the-art methods. The source code and the supplementary materials are available at: https://github.com/MLDMXM2017/CTLMTNet.