Does putting a board under mattress help?

Beside this, can I put a board under my mattress?

For mattresses that are too soft, slipping plywood boards between the mattress and the bed frame or the box spring may help prevent the mattress from sagging or sinking too much. This may give your mattress added support, so you'll be able to sleep better at night.

Beside above, can you use a piece of plywood instead of a box spring? No you don't need one IF you have a foundation for the mattress to lay on, a thick sheet of plywood would do, since that is basically what a box spring us though you bed will lie 4-6 inches lower if you only have a 4 inch typical mattress. So, if you want to buy a firmer mattress, just remove the box spring instead.

Also question is, what can I put under my mattress to make it firmer?

You should use a latex mattress topper or a high-density memory foam mattress topper for a firm yet adjustable support. Latex and memory foam toppers are the best choices to make mattress firmer. If your existing mattress is too soft, you should buy a latex topper, because naturally, latex is firmer than memory foam.

What kind of plywood do you use under a mattress?

The thickness requirements for plywood vary depending on the type of mattress. Plywood thinner than 3/4 inch should not be used, as it will not be strong enough to hold the weight of a bed mattress and a person lying down. The plywood can be exactly 3/4 inch thick, but ideally, it should be thicker.