The coronavirus disease 2019 (COVID-19) pandemic has influenced the everyday life of people around the globe. In general and during lockdown phases, people worldwide use social media network to state their viewpoints and general feelings concerning the pandemic that has hampered their daily lives. Twitter is one of the most commonly used social media platforms, and it showed a massive increase in tweets related to coronavirus, including positive, negative, and neutral tweets, in a minimal period. The researchers move toward the sentiment analysis and analyze the various emotions of the public toward COVID-19 due to the diverse nature of tweets. Meanwhile, people have expressed their feelings regarding the vaccinations' safety and effectiveness on social networking sites such as Twitter. As an advanced step, in this paper, our proposed approach analyzes COVID-19 by focusing on Twitter users who share their opinions on this social media networking site. The proposed approach analyzes collected tweets' sentiments for sentiment classification using various feature sets and classifiers. The early detection of COVID-19 sentiments from collected tweets allow for a better understanding and handling of the pandemic. Tweets are categorized into positive, negative, and neutral sentiment classes. We evaluate the performance of machine learning (ML) and deep learning (DL) classifiers using evaluation metrics (i.e., accuracy, precision, recall, and F1-score). Experiments prove that the proposed approach provides better accuracy of 96.66, 95.22, 94.33, and 93.88% for COVISenti, COVIDSenti_A, COVIDSenti_B, and COVIDSenti_C, respectively, compared to all other methods used in this study as well as compared to the existing approaches and traditional ML and DL algorithms.