If the earth is spinning and we are spinning with the earth, why doesn't it take longer to go opposite the rotation? Imagine you are on a walking type platform at an airport where you are walking with the platform where it moves you faster walking with......than against.  This would be the same concept as if we were walking on earth as we are walking with, or against the rotation of the earth..........or so it seems as the earth and the air are all moving with the rotation of the earth.  
Why can't we walk to our destination faster and with less effort walking WITH the rotation of the earth and why doesn't it take more effort as it would on the walking platform to walk against the rotation of the earth?  Same with driving long distances.  Why is it the same distance taking the same amount of fuel while walking on the platform takes more energy and speed walking the opposite direction?  
 A: When you are walking on a train, why does it not take you longer to walk from front to back than the reverse? Why does it not take more effort? It's because you "share" the speed with the train. At that moment, your reference frame is the train, and only your movement relative to the train affects you. The same applies when you are walking on the earth. Only your movement relative to the earth is important.
If it would be more difficult to walk "against" the earth's rotation (a speed of $1600 km/h$ at the equator), then what about the speed of the earth around the sun ($107,000 km/h$, or $29 km/s$)? It would be harder to walk in some direction in summer than in winter.
Galileo was the first to understand that all movement is relative. He used the example of birds in a ship's cabin to show how the ship does not affect the birds' flight. 
A: Imagine there's an alien up in space watching you on earth, and you are travelling on a high speed train across eurasia. If you travelled from west to east, it would look to the alien that you were moving faster than if you were travelling east to west - the former is going 'with' the travellator, the latter 'against' it. You would also notice a difference - your da (the time between sunrise and sunset) would be shorter going west to east, and longer going east to west.
However, you don't notice it on earth, because eerything is on the travellator - the roads, the oceans, the ground, even the atmosphere.
A: The problem is you don’t specify what your destination is. If you were to start traveling east at noon when the sun was straight up and circled the earth until the sun was straight up again, it would be faster than going the other way. 
If your destinations was the same location on earth then it wouldn’t matter which direction you went unless of course you take wind currents into consideration.
