Interesting challenge! Here's what comes to mind:
The Board Camera: if your game is Go, a camera placed 3 feet above the board should give a relatively accurate and undistorted overview of the game. If your game is Chess, with 2.25" squares, and kings that range from 3.75-4.25", then near the edges, the tops of the taller pieces could appear to cross the line of an adjacent square that, from the camera's perspective, are "behind" the piece. You must decide whether having all the outer pieces appearing to tilt up to 17 degrees away from the camera looks distracting. If you can get the camera 6' above the board, this perspective distortion is cut in half and each piece will appear to occupy only its own square. And if you want to fix the perspective distortion this way, you'll likely need a telephoto lens to fill your frame with an 18"x18" board.
Placing a camera high above a board requires some kind of support platform. If you cannot screw things into your ceiling, a C-stand, a boom arm, and a heavy sandbag should give you a stable platform where you can then mount the camera using an adjustable ball head. This Matthews platform may be overkill, but don't underestimate how much support you need when lever arms get long.
Good external microphones can pick up speech clearly to about 3', though 18" is better for them. You can likely rig two microphones from the aforementioned camera platform so that they are close enough to the players and out of the field of view of the board camera. The cost of such grip equipment might bother you at first, but if you don't get good sound, you will quickly regret all the other money you spent on the rest of your gear.
I think you definitely want to shoot the board in FullHD (1080p) because when action is limited to a quarter of the board, you'll want to digitally crop in to see more detail. A 2:1 digital zoom, which shows one quadrant of the board, would have 540p resolution--better than SD, and good enough for the YouTube crowd.
I know you didn't ask about lighting, but it does inform your camera choices in multiple ways. In the cinema world, there's the "180 degree" rule, which basically says you should keep all your cameras on the same side of an imaginary 180 degree line. Doing so ensures that the cameras see the same key lighting (which is what makes faces pretty, or at least visible) and the same rim lighting (which is what makes profiles pop from the background). You probably won't start out worrying about 3-point lighting, but if you have any success at all, it's the one thing that can make your hundred dollar cameras look like thousand dollar cameras, and your thousand dollar cameras look like $10,000 cameras. Keeping that in mind, you want your side-view cameras to all be on the same side.
If you are the sole camera operator, you can manually cover the rest of the game from a tripod with a fluid head. Speaking from experience, whenever one manually follows the action, action inevitably happens somewhere else, or in a slightly wider or more narrow view than the one selected. Additional static cameras can provide security against that. A static wide camera gives you the freedom to roam with a view toward details. A pair of static cameras looking at the players from across the board gives you the freedom to focus on actual hands-on game play. Etc.
Now, for camera suggestions, I would recommend getting an actual camcorder rather than a photo camera with video functions. The latter are restricted, for tax reasons, to 29 minutes of continuous recordings. Unless you are recording speed matches, you'll want longer continuous run times. I also recommend getting cameras that are more similar to one another than different. The board camera probably needs to be the highest resolution, but the cameras that have humans in the view need the best color (because our eyes will tolerate wrongly-colored game pieces, but not wrongly colored hands and faces). In the end, they probably need to both be about the same level of quality.
Now it comes down to numbers. I cannot make a more specific recommendation without knowing the budget, don't scrimp on sound, lighting, and grip. Overspending on cameras and underspending on other stuff is like buying a car without being able to afford gas or maintenance. Hopefully the above considerations give you a much better picture of how to narrow down your selections and execute a plan.
