Stainless Steel polishing Question

I have a beautiful stainless steel watch that was given to me as a
gift. However, I wear it daily and it’s getting scratched up (as I
knew it would!)

But I was wondering what would be the best way to remove the
scratches and shine it up again. I do have some green rouge
compound, but I’m wary as to what to use to get out the scratches.
Should I go with a fine grit paper, or would a rubber wheel be best
and just get the areas that have the scratching?

My concerns are for protecting the glass on the face and of course
maintaing the smooth & slightly rounded shape on the links.

Thanks,
Julia

On Stainless steel I use a product call Zam with a felt wheel you
can get a mirror finish real easy this way

Rodney Abel