Preventing stains should be your top priority. Marble polishing will eventually come into the picture but you will have an easier job if you would only keep that surface as shiny and as clean as when it came new. Marble is highly porous which means stains will set into the stone as opposed to glass where all you need to do is loosen up the surface dirt. Immediately wipe spilled juice, sauces and other fluids before they can seep into the surface. Keep the surfaces free of debris as these will accumulate and make cleaning more tedious. Avoid damage to the relatively soft surface, meaning don't drag your sofa over that pricey marble floor. Avoid putting hot objects like pans over bare marble surfaces as these will definitely leave a mark. Water can stain you marble shower so spend a few minutes drying the surface to remove standing droplets.
There is cultured marble and there is the real thing. Although cultured or commercially manufactured marble has a hardened surface, it will still need ample care and marble polishing. Try to feel the temperature of your marble surface. Natural marble will feel a bit colder than the air around it while cultured marble will mirror the temperature of the ambient air. Just because cultured marble is tougher than its naturally occurring version doesn't mean you can chop vegetables on that counter. Use cutting boards and your protective pads as if you are dealing with natural marble; the same goes for wiping spills and avoiding hard impact on the surfaces. Absolutely do not use metal polishers on marble surfaces, natural or cultured.
Marble polishing is intended to reduce the scratches on the surface of marble surfaces by restoring the smoothness of the stone. This means it is a good idea to start with an already clean surface. Surface grime is easy. Use ordinary detergent and water to remove the film of dirt covering the marble. Household cleanser will work too as long as you don't use the abrasive kind. Remember, do not use anything abrasive and nothing acidic. Acids will attack natural marble so vinegar is out of the question. There are commercially available cleaners from hardware stores but get only those that are reputable and hopefully from a source that you can trust. There is not much you can do for damaged marble and repairs can be time consuming and expensive.
Marble polishing can already be done when you have sufficiently cleaned the marble. Chemical strippers are used to remove the sealant previously applied to the surface to make it resistant to staining. This will be reapplied again after sanding the marble. Powdered chemical polishers are used on the surface to get a deep glossy finish and then the surface is treated with a sealant to saturate the pores and prevent early staining. Some previous stains will be deep set and you may consider using poultices made from baking soda and water to lift these stains. The ingredients are mixed to a pasty constitution and usually left on the stained surface for around 24 hours. There are also commercially available poultices if you are not comfortable cooking up your own concoction.
