Social media (and the world at large) have been awash with news of the COVID-19 pandemic. With the passage of time, news and awareness about COVID-19 spread like the pandemic itself, with an explosion of messages, updates, videos, and posts. Mass hysteria manifest as another concern in addition to the health risk that COVID-19 presented. Predictably, public panic soon followed, mostly due to misconceptions, a lack of information, or sometimes outright misinformation about COVID-19 and its impacts. It is thus timely and important to conduct an ex post facto assessment of the early information flows during the pandemic on social media, as well as a case study of evolving public opinion on social media which is of general interest. This study aims to inform policy that can be applied to social media platforms; for example, determining what degree of moderation is necessary to curtail misinformation on social media. This study also analyzes views concerning COVID-19 by focusing on people who interact and share social media on Twitter. As a platform for our experiments, we present a new large-scale sentiment data set COVIDSENTI, which consists of 90 000 COVID-19-related tweets collected in the early stages of the pandemic, from February to March 2020. The tweets have been labeled into positive, negative, and neutral sentiment classes. We analyzed the collected tweets for sentiment classification using different sets of features and classifiers. Negative opinion played an important role in conditioning public sentiment, for instance, we observed that people favored lockdown earlier in the pandemic; however, as expected, sentiment shifted by mid-March. Our study supports the view that there is a need to develop a proactive and agile public health presence to combat the spread of negative sentiment on social media following a pandemic.