Scotchbrite pads. They make different pads with different aggressiveness - I would have said "grit" but I don't know if that's the correct term for a pad. In any event, pick up an assortment and start with the most aggressive pad first and work your way down like you would if you were sanding wood. As long as the scratches (why do they stick an "e" before the last s when it's plural? makes no sense) aren't deep, the pads will work fine and give you a new looking sink.
The maroon pads are probably a good one to start with, then green, then white (could probably omit this step). A lot of elbow grease,but it's just one sink so I don't think I'd break out the power tools and flap wheels.
R