ISpeech emotion recognition (SER) is a challenging task due to the limited availability of real-world labeled datasets. Since it is easier to find unlabeled data, the use of self-supervised learning (SSL) has become an attractive alternative. This study proposes new pre-text tasks for SSL to improve SER. While our target application is SER, the proposed pre-text tasks includes audiovisual formulations, leveraging the relationship between acoustic and facial features. Our proposed approach introduces three new unimodal and multimodal pre-text tasks that are carefully designed to learn better representations for predicting emotional cues from speech. Task 1 predicts energy variations (high or low) from a speech sequence. Task 2 uses speech features to predict facial activation (high or low) based on facial landmark movements. Task 3 performs a multi-class emotion recognition task on emotional labels obtained from combinations of action units (AUs) detected across a video sequence. We pre-train a network with 60.92 hours of unlabeled data, fine-tuning the model for the downstream SER task. The results on the CREMA-D dataset show that the model pre-trained on the proposed domain-specific pre-text tasks significantly improves the precision (up to 5.1%), recall (up to 4.5%), and F1-scores (up to 4.9%) of our SER system.