Cleaning it is simple, but If you really want to make it shine ... get yourself a scouring sponge, the ones that are yellow with a green pad on one side are good. Squirt some cream cleaner all round the sink. Then start scrubbing with the scourer, it will take plenty effort but dont worry. When you think you gave scrubbed every edge and corner, get a wet microfibre cloth and wipe all the remains of the now soapy cream cleaner away. Rinse the cloth in fresh water and wipe the sink until it is as clean and dry as you can get it.
To make it shine get a handful of dry paper towel and wipe it as dry as you can, then get another hanful and polish it until it shines.
Take it from someone who is lifelong commercial cleaner ... it works every time, I know it does, she trained me how to do it !