An object sat on the surface of the earth experiences a 9.8m/s^2
True, but it also experiences a 9.8m/s^2 acceleration downwards due to gravity. The two balance out and so the object doesn't move.
The fact it doesn't appear to move due to this acceleration is because we are accelerating
No. We aren't accelerating and neither is the object. If we were accelerating we'd be moving.
There are two opposing forces on the object. The earth's gravity trying to pull it towards the centre of the earth and the force of the ground pushing it up by exactly the opposite amount. If the two forces aren't balanced, the object will be moving.
The fact that we stay the same distance from the centre of the earth is because
space is curved.
No. The curvature of space around earth is caused by the earth's mass and the result of the curvature is what we perceive as gravity.
An object in free-fall has zero acceleration
That is a contradiction. An object in free fall is, by definition, accelerating.