I use a baking powder scrub for counter tops, sinks and toilets - it works great and it is a natural substance. What goes down the drain helps deodorize, especially in the kitchen where food can sometimes get down in the disposal area and create odors. Same with hair and soap scum in the bathroom.
Cleaning rags just have to be washed clean - they don't have to have stains removed. If they are too gross to save, they can be donated in any of the many textile recycling programs going on all over the country. (That's also where I donate stained or ripped clothing, shoes with holes, old pocketbooks, etc.) We have about 3 a year in my town, sponsored by the Scouts or other organizations as fundraisers.
Peroxide based laundry sprays and products, and other non-chlorine products, work great especially when applied promptly. I keep a basin on top of my washer with some water and safe product in it, and throw stained items in there to soak until I get around to doing a wash. If I have a lot, I run a few inches of water right in the washer with a stain remover, and let the items soak until it's time to do wash.
I also use vinegar and hydrogen peroxide for different things. (Don't mix them, of course.)
The problem with chlorine bleach is that its vapors are harmful to breathe and it burns the skin. I've had bleach burns through an accident and they aren't fun. If I have to ventilate the house, use gloves, and wear eye protection, I'm not interested in the product.
I think the key to stain removal is quick action. For sinks, I often run a used washcloth or a rag around it with some baking soda while I'm brushing my teeth with the other hand! Keeps me brushing for the full recommended 2 minutes (a bonus since I sometimes rush that) and the stains don't set, and then the washcloth goes right in the hamper. I'm not sure from your question what it is that you are putting in the sink that's staining it so badly to begin with.
Moreover, excessive disinfecting of common bacteria is a huge problem with the rise of Super-Bugs, the resistant bacteria that create far worse diseases than the ones that got killed off by products like Clorox and Lysol.
ETA: Okay thanks - I was thinking the bathroom sink! Yes, berries and so on stain my white kitchen sink occasionally, although I rinse them in a colander and try to keep them off the sink itself. Same with occasional rust circles if someone's soaking a tin can before recycling. I also get dark marks if I'm scrubbing a metal pan in the sink. I gotta tell you that baking soda takes care of everything with a little scrubbing and a whole lot less money. Just do that spot if you need to, and then scrub the whole sink every 10 days and rinse with hot water. I use an old scrub pad (like a plastic covered one, like Dobie, or I use the reusable/washable towels that are sold instead of paper towels, then throw them in the washer. I use an old toothbrush around the fixtures for any gunk that accumulates. My sink is over 20 years old and still gets plenty white.