Best way to clean silverware?

Another... Before polishing, try washing tarnished silver with warm water and gentle dishwashing liquid and buffing dry with a soft cloth; for light tarnish, this might be sufficient. A frequently recommended, nontoxic trick is to fill an aluminum pan (or one lined with aluminum foil) with hot water, add salt and baking soda, and stir to dissolve. When you add the silver pieces, a chemical reaction occurs, removing tarnish. It's important to note that with this technique, the good tarnish (a desirable patina and the dark crevices in a pattern) may be removed as well, and pitting may result. Experts recommend using a good-quality commercial polish, and there are some less-toxic ones available.

For silver plate:
Take a plastic tub. Put a piece of aluminium foil on the bottom, place the silver pieces to be cleaned on it, so they keep contact with the foil. Sprinkle with pure table salt, pour on boiling hot water. The black stains are silver sufate. The sulfur disappears instantly as a gas (you can smell it). The silver settles back nicely in the cutlery. Nor rubbing needed. Let sit for a bit, and dry with a soft cloth.

I use toothpaste to clean jewelry (all metals) and silver as well. But for silverware the easiest method is the aluminum foil and baking soda trick - I never remember the proportions but you can google it - that's what I always do anyway. You just dump it in the sink and let it soak and it does a darn good job of getting even heavily oxidized tarnish off if you leave it long enough. Or repeat the process. Then you just give it a good rinse. For the rusty bits though I suspect you'll have to do some scrubbing. Good luck!

Each and every time you polish silver with anything but Tiffany silver polish (which is very expensive and I don't know if it is still even available) you will be taking some silver off your piece. Some museums no longer regularly polish silver, .925 or silver-plate, for this reason. All silver polishes are to some extent abrasives, much like the cleanser you use for your sink. The least abrasive, in my experience, which is fairly extensive, is 3M silver polish. It is available in some stores or online. If your piece is rusted and was once silver-plate most likely you have reached the base metal, maybe carbon steel, and your piece will have to be re-silver plated. If it is a sterling silver knife and the handle is in good shape but the blade is rusted, you have a quite old piece of silver. Carbon steel was very often used for the blade of knives since .925 silver is quite soft, doesn't cut all that well.

Wright's Silver Polish has been my go-to forever. Toothpaste is abrasive and will scratch the silver, especially if it is silver plate. When we lived in San Francisco, a friend owned a yacht support business. She employed Georgia, an heir to the Wright business for a while.

Toothpaste has been working magically for me.

I think Wright's silver polish is the best.

Since you are saying there's rust, can we assume this is silver-plate?

Oxidation on silver can be cleaned off with lemon juice and baking soda, and there are commercial silver polishes on the market.

Rust, on the other hand is going to take more work. You can remove it (with sufficient elbow grease) but you will probably have to have that piece re-plated.

I just picked up some silverware as well. I was thinking of using jewelry cleaner on it (because I have some lying around) but I haven't tried yet. Maybe that's a really bad idea to use on something that will touch food but I assume the chemicals will wash off when I've finished cleaning. I'm interested to see what others say on this topic.