For cleaning, a soft wet cloth rung tightly and toweled off, followed by a microfiber cloth for the display (but really, works on the whole Watch).
For polishing, there are any number of fine polishes for stainless steel. I'm not familiar with a 'jeweler's polishing cloth'. I don't know if it will do anything for fine scratches.
For removing fine scratches, I'd look for a very fine polishing compound such as Mother's Mag Polish (which is what I'm going to try). I'll put a minuscule amount worked into smooth cotton towel and see how it goes. By working it into the cloth (maybe this is how a jeweler's polishing cloth works?) first and proceeding carefully, you can avoid getting any polish into the seams between the case and the display and sensor assembly on the back. Very important.