Nicks and dings on watches are pretty rough on new collectors. Your first ding is like your first scratch on a new car: it’s scary, saddening, and you want to just take it back. Now you can.
The Bergeron Scratch Brush costs about $30 (here it is on PMWF) and it can take nicks out of brushed surfaces with a minium of fuss. Remember: this is for brushed surfaces only, so don’t try it on highly polished watches.
Otto Frei has a number of these as well, including smaller brushes for other surfaces. For polishing high gloss surfaces (and I encourage you not to do this with a Dremel and instead try polishing by hand if at all) you need some of this gear and some rouge. Just be careful.