Put some sandpaper on a random orbital sander and have at it. Start with a 120, maybe a 320, and finish with a red Scotchbrite. You can't screw it up.

The cooks kitchen, that's a good idea. I didn't think of that.

Stainless steel sinks scratch. That is just what they do. People who don’t like to see scratches choose a different sink material. As Cook’s Kitchen mentioned, eventually there are so many scratches you don’t see them any more. The first few are hard though.

Keeping it clean with a good all purpose cream type cleanser and buffing it as part of cleaning will render it to the condition where stainless "is what it is" I keep soft rag or two and the specific cleanser in a size container I prefer very close under the sink and do this frequently. Don't like dirty and dull but stainless will not stay scratch free or super shiny. It becomes satiny...but "keeping it up" is a nice gesture for this working area of the space.

Deanna - been there, done that. :) I went a little nutso when I first moved into my pretty new home, trying to keep that stainless sink shiny.

I battled water spots and scratches for about a year.

The transition from new/pretty/shiny to a nice working patina took longer because I was locked in a battle with the effects of life on a sink. Once I finally stopped caring, patina took over. It's easier to love patina than it is to keep it shiny. Your mileage may vary of course. :)