Consumption of sugar-sweetened beverages (SSBs) contributes to the risk of developing overweight and obesity in children and adults. Thus, reducing free sugar is a globally recognized public health priority. The implementation of a sugar tax is one way of achieving this goal. This study aimed to investigate the relationship between familiarity with the sugar tax and its importance in reducing obesity, the perceived health consequences of SSBs, situations and reasons for consumption, and the frequency of SSB consumption. A cross-sectional study was conducted in May 2022 among 216 Polish adults (age 18–45). Hierarchical cluster analysis was used to identify three homogeneous clusters with regard to the perceived health consequences of consuming SSBs. A logistic regression model was used to verify the relationship between the frequency of SSB consumption (dependent variable) and other variables. Respondents with higher familiarity with the sugar tax (OR: 0.594; 95% CI: 0.42–0.85) and those in the “Unaware of disease” (OR: 0.437; 95% CI: 0.18–0.94) and “Disease-conscious” (OR: 0.484; 95% CI: 0.24–0.97) clusters were less likely to consume SSBs at least once per week than those in the “Moderately disease-conscious” cluster. Both the place of SSB consumption (i.e., restaurant—OR: 1.847; 95% CI: 1.14–3.64, work or university—OR: 3.217; 95% CI: 1.51–6.88, family home—OR: 2.877; 95% CI: 1.50–5.53) as well as a preference for their taste (OR: 4.54; 95% CI: 1.66–12.45) increased the chance of consuming SSBs at least once a week. In conclusion, it is necessary to continue educating the public about the health consequences of SSB consumption. The use of legislative measures (i.e., sugar tax) also contributes to reducing SSB consumption and can thus counteract the increasing obesity rate.