Which tool is most suitable for cutting thick cardboard with ease?
The tool you are looking for is called a box cutter which is just a razor blade in a handle. Make sure to put something uncuttable under the cardboard before slicing it with the blade or you will ruin your work surface, tabletop or floors.
What tool is available for keeping under the cardboards by which we can prevent the damage to box cutter and the table?
Cutting mats are avilable in a wide range of sizes. I have A4 and A5, but even A1 (59.4 x 84.1 cm or 23.3 x 33.1 inches) aren't expensive at around £/$/€20. They provide a reasonably non-slip surface (Link is to a UK shop, just happens to have a range on one page).
If you want even bigger, a layer or two of vinyl flooring can work in the short term, but you will get through that with a craft knife eventually, so its best mounted on scrapwood.
Another large, cheap option is hardboard (masonite), but it will blunt the blade faster than a proper cutting mat. MDF is worse still, but either of these can be useful if you have a few very long cuts to make.
