Background: Research on the mental health of students in health disciplines mainly focuses on psychological distress and nursing and medical students. This study aimed to investigate the psychological well-being and distress and related factors among undergraduate students training in eight different health-related tracks in Geneva, Switzerland. Methods: This cross-sectional study used established self-filled scales for anxiety, depression, stress, psychological well-being, and study satisfaction. Descriptive statistics and hierarchical regression analyses were applied. Results: In October 2019, out of 2835 invited students, 915 (32%) completed the survey. Lower academic satisfaction scores were strongly associated with depression (β = −0.26, p < 0.001), anxiety (β = −0.27, p < 0.001), and stress (β = −0.70, p < 0.001), while higher scores were associated with psychological well-being (β = 0.70, p < 0.001). Being female was strongly associated with anxiety and stress but not with depression or psychological well-being. Increased age was associated with enhanced psychological well-being. The nature of the academic training had a lesser impact on mental health and the academic year had none. Conclusion: Academic satisfaction strongly predicts depression, anxiety, stress, and psychological well-being. Training institutions should address the underlying factors that can improve students’ satisfaction with their studies while ensuring that they have access to psychosocial services that help them cope with mental distress and enhance their psychological well-being.