3. Pour a small amount of vinegar onto the sponge and polish water spots. Vinegar also should be used to eliminate any gunk on the sink that was too tough for the baking soda treatment. Rinse the sink basin when finished.

4. Use a kitchen towel to dry the sink. Pour a small amount of oil onto a soft, clean rag. Polish the sink with the oil until it shines, focusing particularly on dull spots.

5. Repeat this process about one a month, or as often as necessary.

Repairing Scratches in Stainless Steel klik in here

For tougher jobs:
Beyond-steel suggests mixing 3 parts cream of tartar with 1 part hydrogen peroxide and then using a damp cloth, massage the mix into the surface. Let dry then wipe with a damp cloth.
1 part vinegar, 1 part hot water, 1/2 part baking soda. Scrub mixture into surface using a damp sponge or cloth. Rinse well then dry with a cloth.
I’ve also been told a tip to spray a layer of hot water over the sink to cover, then spray a layer of household ammonia. Cover with a plastic bag (stretched across the top with ends taped down on counter top) and leave overnight. Rinse the ammonia off in the morning then wash with hot, soapy water and dry with a cloth.

To make it shiny
Wash the surface well then rub in baby oil, or olive oil, or windex and buff dry with a soft cloth

Good scrubbers to use
Soft nylon bristle scrubbers or nylon pot scrubbers. Don’t use anything too abrasive that will scratch the stainless steel.