How can I get stains off of marble?

Try some lime away- http://www.hometalk.com/diy/clean/house/q-how-can-i-safely-remove-severe-lime-scale-form-marble-shower-tile-1858692

But make sure to avoid vinegar and other acidic substances, I usually love using them but they will scratch marble surfaces

I repair pottery and porcelain. When we have stains, we use peroxide - not the kind you by at the grocery store, but stronger stuff you might find through a hairdresser (not creme) or swimming pool store. If you can get your hands on some, you try wetting cotton with then the peroxide, covering the stain with this and leaving it covered on the stone for several days. I've not tried this myself, but I've seen it work wonders on porcelain.

Before start anything, first you should know what types of stains you have? If you get brown stains like heat stain or water stain, in this cases marble poultice would be perfect idea. The most effective solvent for rust stains are a liquid commercial rust remover, but if you have water stains you can easily remove with a marble cleaner and a soft towel. Although there are many options for cleaning it, but there are a chance of damage, so it is advisable for this work one should hire commercial cleaner as professional cleaners quickly assess your needs and will deliver the work within the given time frame and on a reasonable rate.