Why is it that flying west takes longer than flying east despite the fact that the Earth is rotating eastward around its own axis? I had always thought flying west would take shorter than flying east but after a few round trips from New York to San Francisco I've found that it's just the opposite. The same when flying from London to Moscow: flying east is always quicker.
The Jet Stream is the cause for slower flight times to the West. It flows from West to East at high altitudes and therefore helps planes flying East but causes extra resistance for planes flying West.
I had a similar misconception some time ago. Because the earth is rotating it seems that if you go counter to the rotation the trip will be faster. Unfortunately, All the earth is rotating, including the plane you may ride in, if you fly east or west you have to accelerate the same amount to go the same speed east or west, all other influences being equal. In the case of travel, at least in the Northern hemisphere all things are not equal because of the jet stream that Julian mentioned in the previous answer.
I specifically mentioned east west motion because when you travel east to west your relative speed based on your location does not change, but if you travel north or south the surface speed of the earth's rotation changes with latitude. The rotation speed at the equator is just over 1600 kph, but at 45 deg N it is just under 1200 kph, and at 90 deg N it is 0 kph. The change is gradual and for the most part is not noticeable to most people, but it has significant effects when looking at the large air movements of storms. The Coriolis effect, is what causes the low pressure storms in the northern hemisphere to rotate counter clockwise.
See Coriolis Force Wikipeda Skip to the Applied to Earth Section.