What can I do to improve the looks of an old and stained kitchen sink?

Ben, we can help more if we know what it is made of.

What is the sink made of?

Hi, Try using Barkeeper's friend on it

Hi Ben, depending on what the sink is made of, an intense cleaning followed by bleach if it is white will clean most of the stains. Try mixing bleach and water 50/50 ratio and saturating a cloth in the mixture and laying the cloth into and on the sides of the sink. Let soak well and maybe add even more of the mixture as it drys. That should eventually remove any stains. If it is a stainless steel sink you might try scrubbing all the surface with a soap-infused Brillo pad. Hope this helps!

Make a thick paste of baking soda and hot water. Apply to your dampened sink surface. Let set 30 minutes. Re-wet the surface and scrub with a soft cloth, following the grain of the sink. Rinse with very hot water.

Once clean, apply a good car wax to sink surface and buff with soft cloth.

Its a metal porcelain sink

This is what I have. I live in a rental now so replacement isn't so easy