Can modifiers such as "Subdivision" and "Array" be "animated"? Suppose I wanted to animate an object growing out of the ground, or turning from a cube to a much smoother shape, could I animate this using modifiers and key frames?

I know I can use shape keys to achieve something similar, but modifiers would make it easier and smoother. Is this possible?
You sure can! You can animate any of the values in the modifiers by Right Clicking on the field and choosing Insert Keyframe. Then insert another frame further down the timeline where you wish.



You can also insert keyframes from a value by hovering your mouse on it and pressing I, or you can remove keyframes on the current frame by pressing Alt + I.