After take off, does airplane's speed include Earth's movement/speed? Do airplanes turn with Earth movement/rotation?
Aircraft in fact move in the medium of the air.
While they do figure both airspeed and ground speed the former matters to their flight characteristics and the latter to whether you land on time or not.
The air is--taken in bulk--moving along with the rotation of the planet, and winds are the differences between the motion of any particular element of the air and that average.
(All the plus-or-minus's are vector addition or subtraction, by the way.)
Let me try to clarify what the previous posters said.
There are many way to measure an aircaft's speed. The obvious one is the speed relative to the ground. However, if the aircraft is flying into a headwind, its speed relative to the air will be greater than its goundspeed. Similarly, with a tailwind its airspeed is smaller than the groundspeed. As an example, if the plane's airspeed is 1000 km/h with a 100 km/h tailwind, its groundspeed is 1100 km/h. If it was flying into a 1000 km/h headwind, it would be standing still relative to the ground.
There are many more ways to measure the speed though. The earth rotates, which can add or subtract up to 1600 km/h to the speed, relative to the sun. The earth moves around the sun at about 30 km/s or 107,300 km/h. So, if that same plane is flying exactly in the direction that the earth is travelling, it's speed in solar orbit is 108,400 km/h. If it were going exactly opposite to the earth, the speed would be 106,200 km/h. In all other directions the aircraft's speed is between these two extremes, and I did not adjust for the earth/s rotation!
The sun also moves around the centre of the galaxy, and our galaxy moves relative to other galaxies (at different speeds for different galaxies). So we can get an infinity of different speeds for our airplane.
As you can see, there is no such thing as a "final" speed. Speed is always relative to something else. You need to specify what you are measuring the speed against, and you can get many, very different answers.
In fact I wondered about this very recently! The answer lies in the atmosphere, the body of air or water is held in the same position relative the earths surface depending on the density by the earth's gravitational pull. Bodies in these medium will maintain the same relationship unless there is turbulence like wind or wave. Otherwise when you jump up in the equator with the surface speed of 1000miles/hour you would land 1 third of a mile away!
Note that is why parking orbits are possible only outside the earths atmosphere.