Some rings have lots of rocks and stones. I do not polish rings with them. I do not work on these because I don't have enough money for them :) any real metal rings that have designs in the metal or are flat will work nicely. The reason we wear jewelry is for the memories and the shine and style and fashion that it adds. Jewelry makes us sand out and makes our look pop. Nothing rains on your parade more then lack luster jewelry. I do agree that tarnish and oxidation can look good, but only in moderation.
Tools and where to get.
A towel or micro fiber cloth nothing special
Get at any store or a paper towel will work
A can of Mothers Mag and Aluminum Polish. This stuff is a miracle I use it on all my metal. It says on the can perfect for all metal and it is.
Most importantly is the jewelry itself.

Step 1: Setup and Use

You don't need mother polish if you have your own favorite brand, but mothers is cheap at Walmart you can get 10oz. For 5$. If you have never bought polish I recommend mothers. In the pictures above are scorched and oxidized rings 2 silver and 1 copper coin rings. Above are pictures and they are before I've polished them, they look bad :( but not for long :).

Step 2: Polishing Time

Put the towel or rag or whatever you are using on your leg or on a soft table. Next Take just a dab of polish. Then put it on the outside of the ring then rub the ring back and forth do this all the way around.

Step 3: Polishing Time Inside

The inside is hard, a buffing wheel that fits in the rings losely is the best, but if you want to do it cheap you can apply polish to the inside and rub the inside with a cloth/towel.

Step 4: Now It's Done

When you finish it should be sparkly like a mirror. Please enjoy this pearl of knowledge and I hope it has inspired you to polish your jewelry and look amazing.