You’ve hit on something really neat here, but you’re also opening up a pretty big can of worms. Just to tackle the motor question, you can use either stepper motors, or some sort of non-hobby servo motors (in general, servo just means feedback controlled).
Stepper motors have no actual feedback, but move in known increments in response to a control signal rather than rotating continuously. Most ink jet printers use stepper motors to position the print heads and advance the paper, so you can get very precise linear motion out of them.
You could also use regular DC motors, and turn them into servos with some form of feedback like a rotary optical encoder attached to the motor, or a linear optical or capacitive encoder attached to the moving head.
Now of course you’re going to need something better than rubber bands to move your laser, but you’ve got the right idea. You might search around for examples of homemade CNC (computer numerical control) machines, especially CNC routers, which are usually two degrees of freedom (x and y). CNCzone.com is a good place to start, they have a nice DIY section.
As for the laser, I haven’t played with one yet myself, but I’m not so sure that even one of the new high-power green laser pointers is going to do much of a job of engraving. I’ve seen the videos of them lighting matches and popping balloons, but you can also pop a balloon with a dull carrot (it was on Mr. Wizard!). You might be able to mark wood/paper with it, but not really remove material from the surface. You might look into a lens to focus the laser to a finer, and more powerful point.
Commercial laser cutters and engravers use infrared lasers, which transfer heat much more efficiently than visible lasers. Incidentally, desktop laser engraver/cutters start at around $7,500.
Please post any more questions, let us know how you’re proceeding, and good luck!