I also like to use a mild hand soap and a soft (old) toothbrush. When I have stubborn scuffs (on stainless) I turn to a cape cod cloth and microfiber to "buff" out any swirls or scuffs on my case and braclet.
I have a 79260p Tudor which has the dna of a 6263 daytona and I use this method to keep it clean and grime free.
I have also purchased on eBay a Cartier cleaning kit which came with a great soft bristle brush, cleaning solution, and a polishing cloth. It is more than what you likely need but it keeps things organized and looks nice.
