[Eliot Ness]

...I just don't want to do more damage since the scratches are not very deep.

When in doubt start off with the least aggressive paper or polish; it is a lot easier to move to a coarser grade (when needed) than to remove the scratches left by too coarse of paper.

In time you will be able to better judge what grade paper is needed, and try to use a sanding block for flat surfaces. Take some before and after pictures for us too.