How to keep stainless steel kitchen sink bright and clean?

http://thethriftycouple.com/2014/07/30/how-to-clean-a-stainless-steel-sink-and-make-it-shine-simple-3-step-solution-with-ingredients-found-around-your-home/

I dry mine after each use, also Barkeepers Friend is a great cleaner.

I use ScothBrite pads, the green ones and elbow grease, since the sinks are buffed from the factory, I scrub in the direction of the buffing and have removed paint, rust, and everything else. Meyers makes a stainless steel cleaner, I have some, but prefer the "Green scratchy pads"

Run water over all surfaces, then sprinkle a light covering of baking soda on it. Then use a cloth ( I use an old dishcloth) dipped in vinegar to scrub it. You won't have to scrub hard as the baking soda and vinegar will cut through all the dull grunge. Then rinse thoroughly. If needed, repeat until it is really clean. Then do on a regular basis. I have a friend who uses rubbing alcohol to clean hers. but I prefer to use the baking soda and vinegar. I have had my steel sink since 1972 and it still looks good. One additional perk to using baking soda and vinegar is that it helps keep the drain clean, too. Once a month I clean the sink and then pour more baking soda and vinegar into the drain, let it work for a bout 10 minutes then pour boiling water down the drain...clean sink and clean drain. Takes only a couple of minutes... I do it while I fill the dishwasher and wipe down the countertops, by the time I have that done it is time to flush the drain with boiling water and I am done.

Use a Mr. Clean eraser

91% isopropyl alcohol is what keeps my stainless steel sink clean. First I clean it with baking soda and vinegar. Let it dry, then go over it with a clean cloth with alcohol on it.

a box of Arm & Hammer baking soda, sprinker some on and scrub off.

try Bar Keeper's friend, it does an amazing job on stainless steel sinks and appliances

There are a variety of commercial grade stainless steel cleaners available. Use a fine grade steel wool if you want to polish it up after a few cleanings. Do you have hard water? This may be adding to the problem.

After cleaning your sink apply a product called gel gloss for bathrooms and kitchens. It works fantastic to keep the shine.

I think your water is probably the problem. We have a double stainless steel sink in our house that we have lived in 46 years and sink could be much older. Ours cleans up with just some powdered cleanser like Comet. Maybe a water softener for your house? (have no idea how much these cost though)

Use BarKeepers cleanser, it won't scratch it, gets out any rust stains etc as well, it is sold with ajax, comet etc in most grocery stores