Many drawing programs have an option "Display-Smooth" that turns polygons into softer curves on the screen so that they look better. But, underneath, you've still got the (segmented) polygons. If you have the option, you should turn off "Smooth", and then follow the direction above to increase your resolution. You need to raise your polygon count for the STL.
Smoothing is great for on-screen presentation, but you need to be seeing what the print is actually going to produce.
There's a very complicated answer (!math!) for the question "How much should I raise it?", but the simplest explanation is that for MAXIMUM smoothness, no edge along a curve should be longer than 1/2 the step size of your printer (layer thickness or XY step size). Example: If your printer can do 0.1mm steps, then your edges should be no more than 0.05mm long (lookup "Nyquist Theorem")
I do realize that if you actually used that setting, you'd create nearly an infinite amount of polygons and the file size will be monstrous.. but you did ask "How can I get the smoothest print?" ?? You have to trade off file size for 'Accuracy around the curve'.
If a cylinder is 0.5mm, I'll use 16 sides (because you basically can't see it ?? )
If a cylinder is 1.0mm, I'll use 32 sides
If a cylinder is 2.00mm, I'll use 64 sides
and so forth.. if you want a 200mm smooth cylinder, you'll be up at 1024 sides or more
Of course, the same thing applies to ANY curve in any axis, like a sphere. The more segments you have, the smoother your end result will be.