How do you deoxidize stainless steel pots?

I bought some mid-range stainless steel cookware (not nonstick, and it has heavy bottoms). Some of the pots have oxidized--there are swirly colors in the bottom, even when they're clean. How do I get rid of this and make them look new again? Also the saute pan has flecks of gray on the bottom from the last time I made something in it. They won't come off with scrubbing so I'm assuming that's the same thing.