The Earth moves at a high rate of speed around the Sun, and the solar system is moving quickly around the Milky Way. How is it that the Earth's atmosphere is not “blown away”?
Your intuition about objects facing a head-wind when traveling quickly only works near the surface of the Earth in Earth's atmosphere. In the atmosphere air molecules must be pushed out of the way.
In space though, there isn't anything to do the blowing. There is no interstellar medium / fluid that could drag / push or otherwise effect the Earth's atmosphere simply because we're moving quickly.
The Earth does lose atmosphere due to several reasons though. First, there is a "solar wind" from the Sun which are high energy charged particles (protons mostly) that strike the upper atmosphere and impart so much energy to molecules in the atmosphere that they're able to escape. There are other means too which are listed in the Wikipedia article on atmospheric escape.
Though I support to some extent @BrandonEnright reply, it also says that your intuition is not completely wrong. But you were not considering the relevant motion. The effect of speed is due to the interacting surrounding medium, and it is the speed relative to that medium that must be considered, rather than some "absolute speed" which I would not care to define.
In this case, the major interacting medium (though not the only cause of atmospheric loss) is the solar wind of charged particles, going from the sun outward, and you have to consider the motion of earth with respect to that wind, or if you prefer the apparent solar wind (but the speed of earth is negligeable: about 30 km/s earth mean orbital speed against between 300 km/s and 800 km/s for the solar wind speed, in nearly orthogonal directions, hence less than 0.5% effect according to Pythagoras). It is very much like the relative wind seen by a sail boat.
Side-note: Though the solar wind refers to the charged particles emitted by the sun, there is also the flux of photons that may be seen as another kind of wind that has indeed been considered for interplanetary sailing (though the problem differs much from sea sailing in other respects; for example sea sailing takes place in two media, water and air, in relative motion, and that is necessary for controling direction of motion). However, this comparison also breaks because relative speed of photons is a constant, independent of the motion of earth or interplanetary sailships. Energy may be impacted, but not at the relevant speeds.
The relative wind on a sailboat can indeed fly light objects off the deck, as the solar wind can fly atmospheric molecules off the planet. However, in the case of Earth, the magnetic field of the planet diverts the solar wind and protects the atmosphere from it to some extent. It acts like a screen on the deck of a ship that would protect from the wind (but would not help efficient sailing).
The loss of its magnetic field (they are fossile remnants of this field) is thought to be one of the possible causes of Mars very thin atmosphere.
I dont think @Brendon is right. Best exemple of an object moving trougth space and being blown away is a comet (blown away part is the tail).
Best answer I think si that the Earth's atmosphere IS blown away but to a lesser level due to its gravitationnal force (Keeping the Earth in one pieces).