Conspiracy theories present significant societal challenges, shaping political behavior, eroding public trust, and disrupting social cohesion. Addressing their impact requires recognizing that conspiracy engagement is not a singular act but a multi-stage process involving distinct cognitive and behavioral transitions. In this study, we investigate this sequential progression, "recognition," "belief," and "action" (demonstrative action and diffusion action), using nationally representative surveys from the United States (N=13,578) and Japan (N=16,693). Applying a Bayesian hierarchical model, we identify the key social, political, and economic factors that drive engagement at each stage, providing a structured framework for understanding the mechanisms underlying conspiracy theory adoption and dissemination. We find that recognition serves as a crucial gateway determining who transitions to belief, and that demonstrative and diffusion actions are shaped by distinct factors. Demonstrative actions are more prevalent among younger, higher-status individuals with strong political alignments, whereas diffusion actions occur across broader demographics, particularly among those engaged with diverse media channels. Our findings further reveal that early-life economic and cultural capital significantly influence the shape of conspiratorial engagement, emphasizing the role of life-course experiences. These insights highlight the necessity of distinguishing between different forms of conspiracy engagement and highlight the importance of targeted interventions that account for structural, cultural, and psychological factors to mitigate their spread and societal impact.