Avoid spending money on fancy cleaning products instead use WD-40. You most likely already have this product sitting around your house so just grab your can spray your sink and wipe away the excess. It will remove watermarks, fingerprints, and other stains from your sink.
Is the grout around your sink stained? If so you can clean it using a bleaching pen or you can avoid this issue all together by darkening the grout with a grout colorant in a gray or dark beige color. Remember though after you color treat your grout you have to seal it to.