For the polished areas, use Cape Cod cloth .. do not use it on the brushed areas or you will polish them and ruin the finish.
3M scotch-brite re-finishing pads for the brushed surfaces.
If you want to go nuts on this, buy a Dremel tool, jewelers rouge, appropriate dremel tips/wheels and go crazy.
There are at least three excellent tutorials on watch finish care here on the forum that I can think of. Search it. Some members even have them linked in their forum signatures.