Social media is an easy-to-access platform providing timely updates about societal trends and events. Discussions regarding epidemic-related events such as infections, symptoms, and social interactions can be crucial for informing policymaking during epidemic outbreaks. In our work, we pioneer exploiting Event Detection (ED) for better preparedness and early warnings of any upcoming epidemic by developing a framework to extract and analyze epidemic-related events from social media posts. To this end, we curate an epidemic event ontology comprising seven disease-agnostic event types and construct a Twitter dataset SPEED with human-annotated events focused on the COVID-19 pandemic. Experimentation reveals how ED models trained on COVID-based SPEED can effectively detect epidemic events for three unseen epidemics of Monkeypox, Zika, and Dengue; while models trained on existing ED datasets fail miserably. Furthermore, we show that reporting sharp increases in the extracted events by our framework can provide warnings 4-9 weeks earlier than the WHO epidemic declaration for Monkeypox. This utility of our framework lays the foundations for better preparedness against emerging epidemics.