If you really have to clean your silver jewelery, you can try using this, but it is not ideal. Put a tiny amount of your toothpaste on a clean cloth. Gently rub the paste on the jewelery and rinse thoroughly. Your finished product will not look flawless, but the tarnish will be removed.
If your gemstone was treated, you should know about it before you purchase it. You will have to treat different gemstones with different kinds of care. You want to avoid unknowingly cleaning the gem with a chemical or solution that could strip off the protective treatment.
Buy a wedding band that is within your budget; you can always purchase a more expensive one later. You can always get the perfect ring on another important occasion, such as a later anniversary. You may even be able to sell your ring back to the jeweler, in exchange for buying the ring you had your eye on all along.