Old sinks with scratches and patches of missing gloss tend to soak up every stain quickly and be very difficult to clean. I was so frustrated with mine I wanted a new one, so I researched on the web and did some experimenting. Here is what I learned:

1. To clean off stains and the dingy look: Spray your sink with Tilex (Tilex will bleach your clothes and towels so be careful). Put paper towels on your sink bottom so they get wet with Tilex and hold the product on the sink surface. Let sit for an hour or overnight. Toss paper towels and rinse well. This leaves my sink amazingly clean and white with NO elbow grease.

2. Grey pot marks and metallic scuffs come off easily with either ceramic cook top cleaner, or automobile rubbing compound. I use microfiber cloth for this.

3. As soon as your sink is really clean, dry it and wax it with car wax. I use McGuire's liquid, but you can use whatever you have. I use old towels out of the rag bag to polish up the wax after it's dried. Waxing will make your sink MUCH easier to keep clean for weeks to come. Reapply as needed. Now that I've tried this, I will always do it.

4. Get in the habit of drying your sink before you walk away I use a damp or dry microfiber cloth for this.

This leaves my sink looking super all the time with very little effort. Hope it helps you too!