Cleaning stainless steel
Bar Keepers Friend. It comes in a cylindrical container and will be in the kitchen cleaner isle of most stores. It's very cheap.
The directions are on the package. Basically, use a kitchen sponge and a small amount of water to make a paste out of BKF. Rub all over the pot, pan, etc. Then rinse and dry.
For stuborn stains, you may need to use a lot of elbow grease.
For even more stuborn stains, you may want to use the rough side of a heavy duty sponge (e.g. a Scotch Brite Scrub Sponge). I wouldn't use this, though, for regular use. Just for very stuborn marks.
I wouldn't use steel wool or anything that abraisive.
I would be very careful about using any abrasive, even one as gentle as BKF, on brushed stainless steel appliance surfaces - it doesn't take much to create an area that looks very different in reflectivity and stands out like a sore thumb.
I've had the best luck with simply cleaning first with Fantastik, Formula 409, or one of the many similar products. This will usually leave some streakiness that is very difficult to avoid. To get rid of it, I follow up with Orange Glo, a product for cleaning cabinets - it leaves a very slight oily film that eliminates the streakiness. A while back I tried a product specifically for stainless steel appliances called "1-Step Stainless Steel Ceaner" by a company called Home Specialties that appears to work on the same principle, i.e., leaving a very slight oily or waxy film behind. It works great, but is a bit pricey.
I've also found that you can bring back the brushed appearance to spots that may be changed by too-aggressive cleaning by careful application of a 320-grit sponge-backed sanding pad - sold in the painting section of hardware stores. Just be sure to be gentle and move in straight lines in the same direction as the existing grain.