The widespread use of social media has contributed to the increase in hate speech and offensive language, impacting people of all ages. This issue is particularly difficult to address when the text is in a code-mixed language. Twitter is commonly used to express opinions in code-mixed language. In this paper, we introduce a novel Multi-Task Transfer Learning (MTTL) framework to detect aggression and offensive language. By focusing on the dual facets of cyberbullying, aggressiveness and offensiveness, our model leverages the MTTL approach to enhance the performance of the model on the aggression and offensive language detection. Results show that our Multi-Task Transfer Learning (MTTL) setup significantly enhances the performance of state-of-the-art pretrained language models, BERT, RoBERTa, and Hing-RoBERTa for Hindi-English code-mixed data from Twitter.