Family medicine residency programs must balance educational and operational requirements while providing residents with consistent exposure to continuity clinics, a central principle of the Clinic First Model. We study the Family Medicine Residency Scheduling problem and develop a binary integer programming (BIP) framework that incorporates Clinic First principles through two criteria: Clinic Time Consistency (CTC), which promotes consistent clinic involvement within resident schedules, and Level Loading (LL), which promotes a balanced distribution of total resident clinic sessions across time periods. To provide a simple and interpretable schedule structure, we integrate the proposed BIP framework with the staggered scheduling framework , in which resident schedules are generated by successive temporal shifts of a common base rotation schedule. We establish a theoretical bound on the loss in CTC resulting from the staggered scheduling framework and derive valid inequalities that strengthen the resulting BIP formulation. Computational experiments based on data from the University of Minnesota Family Medicine Residency program demonstrate substantial computational benefits of the valid inequalities and show that the staggered scheduling framework incurs only a small loss in CTC. Compared with a schedule used in current practice, the proposed approach substantially improves CTC while producing a more balanced distribution of clinic sessions. Comparisons with an existing scheduling objective further demonstrate the benefits of explicitly incorporating CTC and LL.