Unstable periodic orbits and steady states are believed to form the backbone of spatiotemporal chaos and turbulence, yet their computation in thermally driven flows remains scarce for transitional regimes. In this work we compute and characterize a steady state and three families of periodic orbits in two-dimensional Rayleigh-B´enard at Pr = 1, near the transition to chaos. We find that in its route to chaos, the flow hops between several sets of orbits after becoming quasiperiodic. We use Floquet analysis to study the stability of the orbits obtained and characterize their bifurcations, showing how the appearance of primary and secondary frequencies, as well as phase locking mechanisms, are all related to the dynamics of the orbits. We study in detail how the flow shadows the orbits found and determine in which regimes each orbit is dynamically relevant or not. Our analysis also reveals two important insights: (1) all symmetries are broken before the onset of chaos, and (2) this onset does not alter the behavior of the heat transport.