There is a marble restoration kit available for DIY'ers from a company called BrightStone. Of note is that it is probably the easiest way to get the right kind of polishing compounds in a cream base for bringing the shine back on all kinds of hard surfaces. It works of course on marble, but I have found that it will work with the artificial marble sinks as well as on dull old enameled steel sinks.
There is one full kit that includes rough enough material to take scratches out, to be followed up by two different polishing creams that then bring the shine up. The same manufacturer makes a whole line of special products for cleaning and removing stains from marble, as well as a simple polishing cream when the marble is dull but has no serious wear. The simple bottle of refinishing cream is what works great on the enamel sink. These polishing compounds can often be found in hardware stores, instead of having to go to marble shops or rock stores, but can also be purchased on the web.
Also, the FEIN MultiMaster power tool has a special marble finishing set of sanders that are very effective and much quicker than taking a tough job down by hand. This small but powerful tool is good for small areas, not floors.