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 intend to frame all of my boards with 5/8" square battens.These are carefully glued to the boards with woodglue, 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.
Several of the first boards I intend to make have rivers on them. This means, I want to cut the battens where the river leaves the board. To do this, I use a miter saw to cut neat 45-degree angles and carefully measure so my rivers will always line up when I place the boards next to each other. I chose one edge to be 6" from the corner and all of the rivers will be 4" wide, at least at the edge of the board.
The last step before adding the foam inside the frame is adding felt pads to the bottom of the boards to protect the surface of your table. These pads can be found at DIY stores, craft shops, and even drug stores. They are basically small felt circles with adhesive backs. As a word of advice, although four pads - one in each corner - would be enough to protect the table, I add another pair in the middle of the long sides and one in the center. I noticed on an earlier project that if you only use four pads, the weight of the board tends to draw the center down and causes the board to sag - not what we want after all of our careful work so far!