I use 1/2" MDF (Medium Density Firber) boards for all of my game boards. These can be obtained from most DIY stores and are most commonly found in 2x4' pieces, which is exactly what I use. You could also use 2x2' sections (or whatever other dimensions you like - most DIY centers will even cut the boards for you), but I have found the modeling options to be better with the larger boards. Of course, they are harder to store, but I am lucky in the fact that I have a permanent room devoted to my hobby where I can stack such things.
MDF is more resistant to warping than plywood which is one of the reasons I choose it. However, to make it even more resistant, and to protect the layer of foam that I intend to add on top, I frame all of my boards with 3/4" square battens. These are carefully glued to the boards with Liquid Nails and then to make them stronger, I flip the boards over and secure the battens with 1" wood screws. It's a good idea to predrill holes slightly smaller than the diameter of the screws to reduce the chance of splitting the battens.
You can always skip the framing step if you don't transport your boards and just glue a layer of foam to the board, but the wooden frame makes them extremely durable and adds a very professional look to the finished piece. I think the extra effort is worth the results.
I managed to frame out all six boards in an afternoon. To see the full details of my construction process and see where we'll end up you can check out the tutorial on my main site. I'll duplicate some of the info here and add new photos of this project as I go. If there are any questions, I'll be happy to answer them here!