What solution to clean silver and gold contacts?

Silver does not form an oxide layer. In fact it is very difficult to oxidize silver under ambient conditions. The black tarnish is silver sulfide which affects the conductivity. Depending on the amount of sulfur in your local environment, silver can form a mono layer of silver sulfide in a matter of minutes. 

I forgot to mention that you can use a chemical cleaner to remove silver sulfide. These chemical cleaners are acidic formulations that are not supposed to be abrasive but still etch and pit the silver surface on a micro scale and are not really environmentally friendly or good for your health. I would not recommend them.

I did this experiment back in the 90s with silver in upstate New York. We exposed clean silver prepared in ultra high vacuum to ambient atmosphere for various lengths of time and found a monolayer of silver sulfide formed between 5 to 10 minutes. A monolayer should not affect conductivity and it is not visible to the naked eye but it shows how quickly silver sulfide forms. You do not need to be near volcanoes, chicken coops, or sewage treatment plants to have sulfur in the atmosphere. If your silver tarnishes there is sulfur in the air. When the tarnish is visible, thousands of layers of silver sulfide have already formed. 

Alcohol will not remove silver sulfide. you need an abrasive cleaner. The problem is when you use these abrasive cleaners they can pit the silver on a micro scale which makes the silver surface more reactive with sulfur.

All cleaners will leave a layer of something behind. Even alcohol will leave a layer of hydrocarbons on the cleaned surface, that is unavoidable. 

I should have said silver metal does not readily form an oxide layer in air at room temperature. There is some oxidation but it is slight in comparison to sulfide formation.

Perchloroethylene also called tetrachloroethylene is very effective at removing hydrocarbons but not silver sulfide or metal oxides. Also it is a known carcinogen so care should be taken when using it. Although not as effective, ethanol is safer.

@ Terry9
This is a well known electrochemical reaction to remove sulfur from silver and form aluminum sulfide. I have not studied this myself but I believe it is effective. I cannot comment on whether or not micro pits form using this method. If it works for you and you have observed no adverse effects, keep using it.

Sulfur containing compounds can be present on one’s skin. Depending on their chemical composition they can react with a silver surface.