Wet foam, as a typical multiphase soft material, has complex spatial structure. Foam quality (i.e., gas fraction of a foam fluid), one of fundamental structure parameters of a foam system, generally has a significant influence on the mechanical response of the wet foam to the continuous and oscillatory shear. This study shows that the stress level of the wet foam, including the shear stress and the normal stress difference, rises with the foam quality. An exponential link between the yield stress of wet foam and the foam quality is demonstrated. In the oscillatory sinusoidal shear, a frequent fluctuation of the stress curve mainly occurs at the relatively higher strain rate, and the stress state in the foam is still maintained at the end of the oscillatory shear. Further, with the increase of foam quality, the loss modulus decreases when the foam does not yield, while the storage modulus as well as the loss modulus increases as the strain amplitude exceeds a certain value. Additionally, a nonlinear stress response of the wet foam is mainly attributed to the third harmonic component as the strain amplitude increases in the oscillatory shear. In the shear, the average level of bubble pressure in the foam increases with the foam quality, and it fluctuates with the strain owing to the elastic-plastic deformations of the films. Especially, in the oscillatory shear, the average bubble pressure fluctuates more frequently as the strain rate reaches a relatively higher value.