For axles, you need to consider bending as well as twisting of shaft due to driving end weight and driven end torque respectively. So shear stress as well as bending stress is expected. Most certainly there will be thrust or axial load also. So axial stress will also be present. Consider the axle as beam with simple support at bearings with torque, weight and axial force applied at one of the ends.

If only bending moment is there only bending stresses are expected.

Minimizing stresses depends on what you are looking for and this is not clear from your question. To minimize stress you can increase the shaft diameter which will increase the moment of inertia (I) and distance of top/bottom surface of beam from neutral axis i.e. diameter(y) and hence minimizes the stresses (Bending stress = My/I).