Vaccine hesitancy continues to be a main challenge for public health officials during the COVID-19 pandemic. As this hesitancy undermines vaccine campaigns, many researchers have sought to identify its root causes, finding that the increasing volume of anti-vaccine misinformation on social media platforms is a key element of this problem. We explored Twitter as a source of misleading content with the goal of extracting overlapping cultural and political beliefs that motivate the spread of vaccine misinformation. To do this, we have collected a data set of vaccine-related Tweets and annotated them with the help of a team of annotators with a background in communications and journalism. Ultimately we hope this can lead to effective and targeted public health communication strategies for reaching individuals with anti-vaccine beliefs. Moreover, this information helps with developing Machine Learning models to automatically detect vaccine misinformation posts and combat their negative impacts. In this paper, we present Vax-Culture, a novel Twitter COVID-19 dataset consisting of 6373 vaccine-related tweets accompanied by an extensive set of human-provided annotations including vaccine-hesitancy stance, indication of any misinformation in tweets, the entities criticized and supported in each tweet and the communicated message of each tweet. Moreover, we define five baseline tasks including four classification and one sequence generation tasks, and report the results of a set of recent transformer-based models for them. The dataset and code are publicly available at https://github.com/mrzarei5/Vax-Culture.