Quick rundown of the controls (for the PC/Mac/Linux) version:

 Arrow Keys     character movement & menu navigation
 A     Attack/talk/select in menus
 B Open inventory menu/Back out of menus
 G Dash (or "Go!") along corridors/swap spaces with NPCs
 R     Rotate, i.e. allow direction change w/o using a turn. Hold it while using arrow keys.
 D Restrict movement to diagonal (e.g. up+left arrows). Hold while using arrow keys.
 F     Fire equipped slingshot pellets/use monster's special attack after changing into a monster.

All the keys except the arrow movement keys are remappable to your liking, and Joy2Key is reported to work for using a game controller. Here's a good start on mapping keys to a controller. Built in controller support is coming in the next update (Spring 2015)!

Copyright Peculiar-Games and P.A. Casey.