> But the mechanical part of having something move that fast and in all
> directions seems to me to be the difficult part

Not necesselery

http://www.oatleyelectronics.com/lasers.html

Oatley used to sell a simpler kit using a laser and two speaker coil
assemblies.
Mostly only good for Lissajous figures though

> Maybe moving a mirror instead of the laser would be better, but again
> how to move it like that confuses me

www.jaycar.com.au

Stock number KG9098

Steppers would be needed to draw pre-programmed figures/shapes

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