If you have a fish in a fishbowl and you move the bowl, does the fish immediately slam into the side of the bowl? No, the water generally moves with the rest of the bowl. Air is a fluid like water, it's just a lot more compressible. The air is rotating the same speed as the ground more or less.

The whole reason something can fly through the air is because the air is resisting its natural tendency to fall due to gravity.

An object in motion stays in motion. If the ground is rotating in space, an object on the ground will be rotating similarly. Even if the object moves away from the ground a bit, it's still largely rotating similarly. Without anything pushing on an object, it moves in a straight line. Without gravity, objects would naturally fling off into space. Gravity curves space, though, so that the straight line naturally leads back to colliding with the ground (unless you're going fast enough to be in orbit.)

It doesn't take power to keep moving or rotating. It only takes power to accelerate or decelerate.

You can measure the rotation of the Earth using a gyroscope or a pendulum. Pretty much every science museum has a demonstration of Foucault's pendulum.