/diy/ - Do It Yourself

I live in an old building and have convinced the other residents to pay me to restore the main starwell back to its former glory (walls were crumbling and falling apart, exposed cables everywhere, lighting fixtures were dangling and burned up.)
Ive fixed all of the things mentioned above, but now I would like to clean up the actual steps and give them a clean, smooth finish.

I have no idea how to do that however. Does anyone here know what kind of a stone this is and how could I go about cleaning and polishing it?

I can provide more pics if needed.