the vinegar is a great tip. However, regarding your previous method of steel wool... you shouldn't use that on stainless, you can scrub off the oxide layer and then it can rust - I've seen the aftereffects - a stainless stock pot with the bottom all rusted and nasty.
Scotch-brite pads (and similar) come in a "no-scratch" version (usually blue instead of green) that's made for things like stainless for that reason. I use one with the cleaning product "barkeeper's friend" (which costs about 3 cents a metric ton) on stainless and it's absolutely amazing. A very quick scrub and it looks totally new again. Not quite as low-effort as dumping in some vinegar and boiling, but it's usable in more situations.