What you're describing is a medium sized solid wood panel that is likely to be "cupping".
This is a phenomenon that happens with solid wood boards/panels so the most practical solution for a beginner would probably be to use a different material - such as plywood or MDF. These engineered materials will be perfectly flat from the factory, and are not subject to the wood movement forces that a solid board or panel would be subject to. A nice hardwood plywood with edge banding that you add could still look nice for your board gaming application.
What if you want to salvage your existing panel? You're going to have to constrain the cupping while still allowing for seasonal wood movement. There are many approaches to doing this but I will outline just 2:
- Use metal c channel or angle to constrain wood movement on the bottom, or on the sides. A flat strip of metal on the bottom will not resist the cupping action of the panel. Fasten the metal channel/angle to the sides or bottom of the panel using screws in slotted holes to allow for wood movement. To see if the metal you have selected is strong enough for the task, try clamping your cupped panel to metal channel and see if the panel flattens out. If the metal is too weak, you'll notice the metal bending and the wood remaining cupped. See, for example, this large drafting table top kept flat in part by the metal channel on the sides:
- Make breadboard ends for the panel. This is a fairly advanced woodworking technique for keeping solid wood panels flat on tabletops or, originally, breadboards! Ultimately, the goal here is the same as with the metal - use a cross-grain support to constrain the cupping, while still allowing for seasonal expansion/contraction of the panel. There is a plethora of breadboard end information out there - you just need to know the term.
In summary - the easiest solution for a consistently flat wood panel is to use an engineered wood material like plywood or MDF. If you want to use the panel you already have, you may be able to contain the cupping with some strategically placed, cross-grain c channel. For those looking to tackle a more traditional/advanced woodworking solution to the problem - this is a case for breadboard ends.