For something like chert I would check out a rock shop, you can find a slab for cheap I bet. Chert is microcrystaline quartz so flint, agate, heck you name it could work. Jasper too, I think a lot of the good Japanese stones must be jasper. I have a huge rock saw I need to finish setting up so I can cut some slabs myself.
Slate is just that, it is essentially mud that is fossilized or shale as that is known. I could imagine it could make a good slurry for a stone but that is just a guess. It is pretty soft but the cleavage is nice for making flat objects.
I'd consider a piece of agate for a hone, I bet that would be nice.You should be able to find something solid for cheap. I pay 1$ a pound for clean but not good looking agate slices out of the local rock shops random junk" pile, a 1# slab would probably do the trick.
Just guessing..... I know my rocks but am not too sure what makes great sharpening stones.
I do know you can buy slabs of the good arkansas stones but that kind of defeats the purpose. I'll look in my site books and see if I can find you a local source that you can collect from.