This COVID-19 pandemic is so dreadful that it leads to severe anxiety, phobias, and complicated feelings or emotions. Even after vaccination against Coronavirus has been initiated, people feelings have become more diverse and complex, and our goal is to understand and unravel their sentiments in this research using some Deep Learning techniques. Social media is currently the best way to express feelings and emotions, and with the help of it, specifically Twitter, one can have a better idea of what is trending and what is going on in people minds. Our motivation for this research is to understand the sentiment of people regarding the vaccination process, and their diverse thoughts regarding this. In this research, the timeline of the collected tweets was from December 21 to July 21, and contained tweets about the most common vaccines available recently from all across the world. The sentiments of people regarding vaccines of all sorts were assessed by using a Natural Language Processing (NLP) tool named Valence Aware Dictionary for sEntiment Reasoner (VADER). By initializing the sentiment polarities into 3 groups (positive, negative and neutral), the overall scenario was visualized here and our findings came out as 33.96% positive, 17.55% negative and 48.49% neutral responses. Recurrent Neural Network (RNN) oriented architecture such as Long Short-Term Memory (LSTM and Bi-LSTM) is used to assess the performance of the predictive models, with LSTM achieving an accuracy of 90.59% and Bi-LSTM achieving an accuracy of 90.83%. Other performance metrics such as Precision, Recall, F-1 score, and Confusion matrix were also shown to validate our models and findings more effectively. This study will help everyone understand public opinion on the COVID-19 vaccines and impact the aim of eradicating the Coronavirus from our beautiful world.