So I understand how a free-falling object in a gravitational field is "at rest" or has an inertial reference frame, but I cannot figure out why an object hovering in a gravitational field is supposed to be constantly accelerating? Take me, for example, standing on earth. Am I accelerating? Constant acceleration, to me, seems to mean a continually increasing velocity. Is that cancelled by the acceleration of gravity somehow, and can we even think of gravity as an acceleration?
In the context of general relativity when standing 'still' on the ground you are subject to exactly one force: the normal reaction from the ground (which is after all the thing that keeps you from following the natural trajectory for a person in that location), so you are subject to a net force and therefore accelerating.
Of course, this is a big change of viewpoint, and not very useful in many contexts. Luckily we can just use the accelerating frame of reference that remains fixed with respect to the ground if we are willing to introduce the gravitational pseudo-force and thereby recover the point of view taken at the introductory level. This is analogous to introducing the centrifugal, Coriolis and Euler pseudo-forces if we want to use a hull-linked frame of reference in spin-gravity space-station.