I have two models - we'll say a sphere and hollow cylinder - that I'm working with. Effectively, I want to 'subtract' the hollow-cylinder shape from the sphere. I know from certain orientations (the top of the sphere) this is fairly easily done, just a matter of deleting and creating new faces. However, from other orientations the edges/geometry doesn't line up so easily. Is there an automatic (or at least, not terribly difficult/manual) way to 'subtract' one mesh from another?
You can use a Boolean modifier.
For example, adding a boolean modifier to a sphere with the operation set to Difference, the parts of the sphere which intersect the object specified in the modifier will be subtracted.
As a bonus, a quick way to select an object for a modifier (or anywhere a similar selection interface is used) is by pressing E while hovering over the object selection drop down and clicking on the desired object in the 3D view.