Silver jewelry can be polished with any commercial polishing cloth. I don't recommend liquid dipping cleaners for my pieces. It will remove the detail oxidation, the dark areas.
I wash the pieces in Dr. Bronner’s soap with a soft very toothbrush before I use polishing cloths. Sometimes that's all that's needed. I'll use Dr. Bronner's afterwards, too. It gets rid of residue from the polishing cloths. It also makes the gemstones sparkle.
There is also anti-tarnish cloth, which can be expensive.
Any pieces that are entirely black or have a lot of patina on them, I have waxed to keep the black from being damaged.
For the pieces with gem stones I would say be careful. Like take off the rings when you wash dishes. Keep your jewelry away from clorox or chlorine, especially silver. It affects the alloys in the metals. Don't use paper towels to clean as they scratch the metal. Use a very soft cloth or use a toothbrush very gently in the crevices.