Recent drastic increases in natural gas prices have brought into sharp focus the inherent tensions between net-zero transitions, energy security, and affordability. We investigate the impact of different fuel prices on the energy system transition, explicitly accounting for the increasingly coupled power and heating sectors, and also incorporate the emerging hydrogen sector. The aim is to identify low-regret decisions and optimal energy system transitions for different fuel prices. We observe that the evolution of the heating sector is highly sensitive to gas price, whereas the composition of the power sector is not qualitatively impacted by gas prices. We also observe that bioenergy plays an important role in the energy system transition, and the balance between gas prices and biomass prices determines the optimal technology portfolios. The future evolution of the prices of these two resources is highly uncertain, and future energy systems must be resilient to these uncertainties.