Elbow grease works best. Seriously, stainless forms its' own coating with the chrome it exposes to oxygen. I would use some sort of scotchbrite. If you really want a chemical then there's a Canadian engineering firm marketing stainless cleaners and their name is Walter.
I agree, elbow grease is probably best method. There are some commercially available products one can find at Home Depot, Lowes or most home centers but these also require some elbow grease. 1/3 Nitric acid with water could be used but this is likely excessive. Suggest power wire brushing using stainless steel wire brush on the end of a drill motor. Once you get it completely clean, just be careful about the little things, like finger prints, which contain body salts and make lead to restaining. In the end, if you have real difficulty, go to a welding supply house and ask for some pickling paste, that will definately work for you.
Yes, there's a product called Derustit that is a gel of nitric and hydroflouric acid and I'm personally scared to death of it. Combined with a slight electrical DC charge, you can even electropolish SS, but I'm not hip to all that horrible chemical stuff.
What You do to a stainless part to restore the finish depends on what type of finish is on it presently, and I am not talking about coatings.
If it is polished, scotchbright will take the shine right off it and leave it with small hard to remove scratches.
If it is abrasive brush finished [has fine lines going in all the same directions] scotch bright will work, just be sure that the direction You work is parallell to the existing finish.
To remove light staining or rust marks Bar Keeper's Friend, a powdered cleanser, works well on all but the most highly polished items. If there is a directional finish, work parallell to the lay of the finish. You can get the BKF at grocery stores, or a K Mart, Wallmart etc. type stores. This product has oxalic acid in it, and that is why it works.
If You need to buff a polished surface [with a rag wheel on a power buffer] there are compounds specificly for stainless steel. Some compounds won't cut stainless effectively.