Cyberbullying is a serious societal issue widespread on various channels and platforms, particularly social networking sites. Such platforms have proven to be exceptionally fertile grounds for such behavior. The dearth of high-quality training data for multilingual and low-resource scenarios, data that can accurately capture the nuances of social media conversations, often poses a roadblock to this task. This paper attempts to tackle cyberbullying, specifically its two most common manifestations - aggression and offensiveness. We present a novel, manually annotated dataset of a total of 10,000 English and Hindi-English code-mixed tweets, manually annotated for aggression detection and offensive language detection tasks. Our annotations are supported by inter-annotator agreement scores of 0.67 and 0.74 for the two tasks, indicating substantial agreement. We perform comprehensive fine-tuning of pre-trained language models (PTLMs) using this dataset to check its efficacy. Our challenging test sets show that the best models achieve macro F1-scores of 67.87 and 65.45 on the two tasks, respectively. Further, we perform cross-dataset transfer learning to benchmark our dataset against existing aggression and offensive language datasets. We also present a detailed quantitative and qualitative analysis of errors in prediction, and with this paper, we publicly release the novel dataset, code, and models.