When you remove the linoleum/vinyl, take up the underlayment that it's surely attached to. Pound down the fasteners... you don't have to sand the subfloor, but do screw it off to the joists. The TCA would have you install plywood to bring the total thickness to 1-1/8". That underlayment looks like the Custom branded product that HD sells - use thinset that's just a little on the damp side to lay it and have some loose boards handy to hold down the ends until it sets, as they tend to curl up. Apply two coats of sealer to the face of your stone before laying, as it's near impossible to get bonded thinset off slate. Do a tidy job undercutting casings, and use a straightedge to get the face of the stone relatively flush along the walls, or your baseboards will be a challenge.