Here is a photo of my sink before
It's not the easiest to see the stains in this picture but it's the best I could get - but it had stains.
I sprinkled some dishwasher detergent and let sit a minute or so and then scrubbed a bit
Voila! Clean and nearly stain free, for the most part. The instructions did say that if the stains were really bad, to do this as step 1. Step 2 is to then use a bleach spray and then use Bar Keepers Friend. Thankfully, the dish detergent did the trick, as that is the only item out of the 3 that I had.