Twitter is one of the most popular social media applications used by the general public to debate a wide range of topics. It is not surprising that the platform has become an effervescent channel where people are talking about the COVID-19 pandemic. After one year of a severe pandemic, we are now giving the first steps towards its ending: the production and distribution of vaccines as well as the start of vaccination campaigns in several countries worldwide. However, the relatively quick emergence of alternative vaccines raised several concerns and doubts among the general people, leading to lively online and offline debates. In this paper, we investigate the public perception of this topic as it unrolls in the real world, analyzing over 12 million tweets during two months corresponding to the early stages of vaccination in the world. Our investigation includes the analyses of user engagement as well as content properties, including sentiment and psycholinguistic characteristics. In broad terms, our findings offer a first look into the dynamics of the online debate around a topic - COVID-19 vaccination - at its early stages of development, evidencing how people use the online world, notably Twitter, to share their impressions and concerns about it. As a means to allow reproducibility and foster follow-up studies, we release our collected dataset for public use.