The main thing is to sort out the moisture situation before you insulate anything. The previous answer correctly pointed out basement ventilation, but there's another thing to check: make sure water isn't leaking in from outside and condensation isn't building up on the pipes. If the crawl space is damp - that's a real problem, and no insulation will help, it'll only make things worse. But if it's dry and has decent ventilation, then you can start.
As for the materials themselves: mineral wool in floors is risky - it absorbs moisture quickly and loses its properties. Foam or extruded polystyrene (XPS) is way more practical for a ground floor - they don't mind moisture and keep heat stable. Polyurethane is also a good option, but pricier. If you don't want to tear up the floors, there's a workaround with removable wooden floors that have insulation - you lift them up, put the material underneath, and put them back. It costs less than a full overhaul, though it's more labor-intensive.
But honestly, if the floors are really freezing - a half-measure might not cut it. Then you'll have to remove the old flooring and do it properly: vapor barrier, insulation, moisture barrier, new floor. Sounds expensive, but you end up with comfort for years to come.