Cleaning stones?

I have been browsing through the message boards on this website on and off since last night and I still have a couple questions regarding general care of the stone in my "new" apartment.  Of course I haven't read ALL the messages, so please forgive me if I'm repeating issues. 

Two very hectic weeks ago I moved into a new apartment with a LOT of stone surfacing (at least it's a lot for the size of the apartment).  Of course I am NO expert in stone, but here is what I have.  The bathroom floor and walls (including walls in the shower) are what I believe to be marble (dark green/black on floor, green/white/what appears to be a peachy/tan on the walls).  The kitchen is floored in polished granite with a matching backsplash.  All surfaces still have a nice shine on them (the bathroom less so than the kitchen-could that be properties of the stone itself?) and appear in very good condition.  Being in Germany, I have no reason to believe that the materials used are not natural. 

Now, I have had an (albeit ameteur) understanding of the delicacy of  marble with acids/chemicals, since I was 10, so I have NOT used any type of chemical or home remedy on any of my surfaces.  I have been using a squeegee on the shower every day since I moved in so the tiles, for now, are still clean.  The fixtures, however are showing water spots that I will be wanting to clean off soon, and eventually I will need to clean the tiles themselves.  My biggest question is (and forgive me if I'm about to offend anyone's sensibilities):  What is the safest, least expensive and yet easy to come by method of cleaning the water spots off my fixtures/floor pan, and eventually the soap scum from the walls, without harming the finish of the stone and incurring damages when I move out.  The grout used is also a neutral gray color, if that will make a difference.  (Please understand that I am on a single-income military budget AND living overseas where my options are limited by selection at my commissary, stuff that will ship APO, and an economy where I barely understand the language.)  I fully appreciate the look (the fact that I'm not THAT fond of green is neither here nor there) and am willing to put in some elbow grease to keep the surfaces looking nice-for the stone's sake, not just to avoid damage charges.

Also, there are some minor cracks in the grout near where the shower head is mounted, what is the urgency in getting that fixed? 

As for the granite in my kitchen, I know even less about its weaknesses than I do those of marble.  What kind of things do I need to be cautious of with granite?  And what will be the best way to clean the greasy residue from around the stove? 

Any help you can give me would be great. 

Melissa, USA in Deutschland  28Feb