We numerically explore the two-dimensional, incompressible, isothermal flow through a wavy channel, with a focus on how the channel geometry affects the routes to chaos at Reynolds numbers between 150 and 1000. We find that (i) the period-doubling route arises in a symmetric channel, (ii) the Ruelle-Takens-Newhouse route arises in an asymmetric channel, and (iii) the type-II intermittency route arises in both asymmetric and semiwavy channels. We also find that the flow through the semiwavy channel evolves from a quasiperiodic torus to an unstable invariant set (chaotic saddle), before eventually settling on a period-1 limit-cycle attractor. This study reveals that laminar channel flow at elevated Reynolds numbers can exhibit a variety of nonlinear dynamics. Specifically, it highlights how breaking the symmetry of a wavy channel can not only influence the critical Reynolds number at which chaos emerges, but also diversify the types of bifurcation encountered en route to chaos itself.