To polish metals retains elements that the user should know to achieve a good finish. The use of sand paper from heavy to light seems usual but it's also useful to know about the different kind of polishing compounds.
Using rotary tools and accessories, sometimes the compound doesn't worth to polish hard metals because the accessory is too soft. So it could polish until a limit, or very shiny with some metals, but maybe not with stainless steel.
Some polishers have a formula that doesn't require special equipment but it's also possible achieve a mirror finish in your home with a rotary tool, silicone based polisher and an accessory designed to polish hard metals. This is, sadly, not the one that you usually find in known names of rotary tools. There's an exception, the Dremel 520 although it's also soft enough to be damaged with the metal and it's too expensive for what it is.
Another must is the white compound, a special kind of wax that provides the mirror finish or something very similar to make you very happy with the result. Of course, professional polishers do a better job since they know well the involved elements. Anyway, the rotary tool way it's enough for some.