Under ambient conditions, water freezes to either hexagonal ice or a hexagonal/cubic composite ice. The presence of hydrophobic guest molecules introduces a competing pathway: gas hydrate formation, with the guests in clathrate cages. Here, the pathways of the phase transitions are sought as sequences of states with coexisting phases, using a generalized replica exchange algorithm designed to sample them in equilibrium, avoiding nonequilibrium processes. For a dilute solution of methane in water under 200 atm, initializing the simulation with the full set of replicas leads to methane trapped in hexagonal/cubic ice, while gradually adding replicas with decreasing enthalpy produces the initial steps of hydrate growth. Once a small amount of hydrate is formed, water rearranges to form empty cages, eventually transforming the remainder of the system to metastable beta ice, a scaffolding for hydrates. It is suggested that configurations with empty cages are reaction intermediates in hydrate formation when more guest molecules are available. Free energy profiles show that methane acts as a catalyst reducing the barrier for beta ice versus hexagonal/cubic ice formation.