Firstly, thinking of gravity as a force is a fairly decent way of understanding it. The model breaks down in some situations, but is not bad most of the time. Gravity as a force is a helpful lie, which is why we teach it.

The GR way of looking at gravity is as curvature in spacetime. Essentially the presence of energy/mass squishes space and time together around it, meaning that there is "more space per space" close to a massive object, and "less time per time" (time passes slower).

One way of thinking about falling is that this effect twists an object's time direction a bit into its space directions. The object falling is sitting where it is doing its normal thing, staying still and going forwards through time at 1 second per second. But from an outside perspective it is going forward through time at a bit less than 1 second per second, and is also moving downwards a bit, because its "forward through time" direction is an outsider's "forward through time and a bit down" direction [disclaimer; this isn't quite how the maths works, but is a helpful analogy].

A more massive object has more of a twisting effect on spacetime, so this is a bigger deal for more massive objects. It's also a bigger deal closer to massive objects than further away; acceleration due to gravity is about 9.81 metres per second-squared near the surface of Earth, but it drops the higher up you get. It is about 90% of that on the International Space Station, for example.

I'm not entirely sure what you are saying about accelerations of objects. It is possible to mimic the effect of gravity by being in an accelerating reference frame - that is kind of what a g-force is (which is not due to gravity, and not a force - great naming there, guys!). But in that case we're simulating the effects of gravity - depending on the acceleration we can vary the fake-gravity's strength.