I have a slate floor - large slate tiles.It got very scratched by removal men when we moved in. I cleaned it with HG polish remover then got down on my hands and knees and used black patination polish. From then on (3 years now) I just mop it and about every three months use HG Golvpolish on it (with a mop). It always looks clean and shiny - it's in the hall and i only mop it about once a fortnight (slob that I am!)
We have a slate floor at work in the loos. I have to be very careful what I use to clean it or it becomes like a skating rink. So far I've only done everyday cleaning on it using Washdown (a commercial product, but it's the same as Flash liquid) diluted in the mop bucket.
I used to use some bleach in the water, but since the order changed to thick bleach I haven't been able to - thick bleach must contain some sort of soapy stuff that made the floor lethal.
What I am looking for is a kind of black slate polish that I can use on the floors and the hearths every couple of months. Haven't really explored yet but I will let you know if I spot anything. Aylsham's tips sound good!