Newbie question

Two really dumb, newbie questions:
How to I turn off snap so that I can freely move an object within an object, and not have it snap to a grid?
And how can I subtract a smaller shape from a larger shape?

Grid Snap can be turned off using Snap command. Rhinoceros Help

You can use here Boolean Difference Boolean Difference

If the smaller shape and the larger shape have been extruded to the same length and on the same plane, then one can select both extruded shapes and use the Trim feature to select trim the smaller shape out of the larger one.