Edit: but... the laser doesn't seem powerful enough to burn anything but black plastic. Stupid.
That's because diode lasers (well, the ones you can generally get without a bank loan) don't have enough power; for a homebrew diode laser cutting system, I wouldn't even think of any laser under 1 watt of power.
Most laser cutters use CO2 water-cooled gas lasers outputting around 25 watts or more. It is possible to build such a laser (although the end mirrors won't be cheap; everything else though is fairly inexpensive), but they aren't small (even commercial tubes are about a foot long or so - homebrew would be larger), so you can't mount the tube and cooling system to the moving gantry. Instead, you must use mirrors to steer the beam. Since the output of such a laser is IR, you can't use just any mirrors, either - you have to use special mirrors that can reflect (instead of absorb) IR. Like the end mirrors for the laser, these mirrors (and lenses) won't be cheap. You could easily spend several hundred dollars on the optics alone.
Then you're left with putting it all together, and aligning/focusing everything so it works. Here's the final problem: You're messing around with a 25 watt, invisible beam of light, to which your eye has no blink response. Any reflection into your eye is going to cause damage. This is really true of any laser over a milliwatt or so, but when you get into the 10's of wattage range and beyond; well, let's just say you're going to want some eye protection. So now you need a pair of wrap-around filter glasses that filter the IR band you are likely in. Those aren't cheap, either.
In the end, you would be better off just purchasing an inexpensive laser cutter off ebay.