Here when I first read this thread topic headline, it was about aftermarket SS vehicle grills, not cooking grills!
I agree with Dan above, the grain(sometimes referred to as brushed) pattern in most appliance SS needs to be "followed" (IE, in the same direction as the grain lines)when wiping with any cleaning media.
I would never use steel wool, even 0000 "grit", but 3M does have some scouring pads that supposedly do not "scratch" SS. As far as having something that will keep SS from tarnishing or oxidizing, I know of not that will withstand the high temps of a grill, even insulated tops. My suggest is to use some type of metal polish for stainless steel, but even that will burn off this first time the grill is used. Just do not use a chrome cleaner as they end to have abrasives that may scratch the SS.
Most sheet metal SS used in appliances is a "cheaper" 400 series type SS, which is why it will tarnish or oxidize. One way to test for cheaper 400-series SS is to use a magnet on the sheet metal. If it is magnetic, it`s a low-grade (AKA cheap) SS. If it were a true 304 food grade or 316 surgical grade type, the cost would be prohibitive. Also, the gauge or thickness of the sheet metal can vary and the cost and quality of the appliance will reflect that. That "fancy" pressed/ embossed logos or names or channels or bent/pressed diamond surfaces in the sheet metal acts as stiffener ribbing and keeps thinner gauged sheet metal from deflecting, or "oil canning or drumming" as it is called in SS sheet metal manufacturing.