Since December 2019, the world has been fighting against the COVID-19 pandemic. This epidemic has revealed a bitter truth that though humans have advanced to unprecedented heights in the last few decades in terms of technology, they are lagging far behind in the fields of medical science and health care. Several institutes and research organizations have stepped up to introduce different vaccines to combat the pandemic. Bangladesh government has also taken steps to provide widespread vaccinations from January 2021. The Bangladeshi netizens are frequently sharing their thoughts, emotions, and experiences about the COVID-19 vaccines and the vaccination process on different social media sites like Facebook, Twitter, etc. This study has analyzed the views and opinions that they have expressed on different social media platforms about the vaccines and the ongoing vaccination program. For performing this study, the reactions of the Bangladeshi netizens on social media have been collected. The Latent Dirichlet Allocation (LDA) model has been used to extract the most common topics expressed by the netizens regarding the vaccines and vaccination process in Bangladesh. Finally, this study has applied different deep learning as well as traditional machine learning algorithms to identify the sentiments and polarity of the opinions of the netizens. The performance of these models has been assessed using a variety of metrics such as accuracy, precision, sensitivity, specificity, and F1-score to identify the best one. Sentiment analysis lessons from these opinions can help the government to prepare itself for the future pandemic.