Marble needs to be cleaned regularly with a  soft cloth or sponge with a solution of mild, liquid dishwashing detergent and warm water. Rinse. Dry with a soft cloth.

Do not use vinegar, lemon, or any other acidic cleaner; it can eat into the marble causing long term damage.

If there are stains on the surface, rub them down with a marble polishing powder, such as tin oxide (available at hardware and home-improvement stores). Carefully follow the product's directions.

Stains on Marble can also be cleaned using baking soda (bicarbonate of soda). As with all projects you must weigh up the value of the item to see whether you can tackle the job yourself or leave it to the professionals. If I were cleaning marble that was not of any great value then I would use a paste of baking soda and water and apply with a soft cloth. Baking soda has an abrasive action that does not actually scratch the surface, making it ideal for marble.

The same would apply for cleaning slate.