The Earth is spinning/moving from west to east (left to right on a map) at 1,000 mph at the equator, 500 mph at 45 degrees north or south latitude, and 0 mph at the poles.
The plane still has this rotational speed, even after it takes off.
In the same way, a space ship still has the rotational speed of the earth as it takes off. That's why it saves fuel to launch rockets into earth orbit from near the equator: they get the maximum (1,000 mph) head start on achieving orbital velocity, about 17,500 mph, I think.
The atmosphere rotates with the earth, more or less. But any air mass moving towards the equator has LESS eastward velocity, and therefore moves west in relation to the surface. Air masses moving away from the equator have MORE eastward velocity than the atmosphere further from the equator, so it moves east. This is called the Coriolis effect. It explains why hurricanes, for example, have the rotation they do.
MRPLUTO