How to combine a Shade Smooth and an Auto-Smooth object together in Blender? Jachym's answer is great, but I would like to add a few additional notes I discovered today:
The result of Auto-Smooth shading will look different than if you simply select certain faces and set Shade Smooth or Shade Flat (via the Object Context Menu, accessible via "W" key for Right-Click Users, or by right-clicking for Left-Click Users).
If that's the look you want, then great. However, if you want to exactly match the shading achieved with Auto-Smooth, you will need to use an Edge-Split Modifier (either with the "Edge Angle" setting, which is identical to Auto-Smooth behavior, or with the "Sharp Edges" setting, where you manually choose what edges will be sharp by pressing Ctrl E > Mark Sharp or Edge > Mark Sharp on selected Edges in Edit-Mode).
The differences in Shading can be clearly seen in the 3 example images below:
1. Faces set to "Shade Smooth", "Auto Smooth" Enabled:
2. Faces set to both Shade Smooth and Shade Flat, "Auto Smooth" Disabled.
Note how the reflection on the inner rim has CHANGED:
3. Faces set to "Shade Smooth", Edges Marked as "Sharp", and EdgeSplit Modifier.
Note how the reflection in the inner rim matches the Auto-Smooth version EXACTLY:
If you use the Edge Split Modifier version, be sure to APPLY the modifier before joining this object to a different object that uses Smooth Shading on all its faces. You can apply any modifier by hovering over it an pressing Ctrl A.