If you're set on using wood and I assume the slots you want are rounds, you need either a forstner bit or a hole bit for a drill. Preferably you'd use a drill press to plunge the bit into the wood and you could control depth while maintaining a perfect 90 degree angle. You can also use a hand drill with either, but you need to be careful about keeping it straight. Make sure the wood piece you're drilling is clamped securely, or your bit may snag and drag it.
Forstner bits are much better, they create less tear away and cut more flush to the wood. They're more like router bits that cut vertically down. Hole bits are just used to drill...holes, but they basically just punch through whatever you're drilling through and aren't designed for a great deal of precision depth. Either comes in a variety of sizes.
Alternatively if you don't want to acquire any specialized tools or bits, you could probably use a good dimensionally stable wood as a base (to prevent warping), and cut out your slots in a piece of foam core sheet, then laminate that on top. Once it's flocked, you probably wont be able to really tell it's foam core.
If you're laminating a metal sheet in there, then you can't use wood glue to adhere it to wood (metal is non-porous); you would need some other type of epoxy. You could also just cut out circles from a metal sheet and glue them into your slots. This would be easier with a soft metal sheet you can cut with scissors with that degree of small precision like...maybe thin brass. Alternatively you could just countersink individual rare earth magnets into each slot, or even iron nail heads or tacks, but I would check that the hold is good enough for your purposes first.
I would also consider weight. Even with just the display board, the pounds are going to add up fast with the wood, metal, etc. Plus the mountain of minis you'll be putting on it. Find some handles in the hardware section that you like and secure them to your board for easy handling.