How is it possible for objects to travel faster than the speed of sound when particles interact at the speed of sound First of all, I am sorry if this is a stupid question but:
I've heard that atoms interact with each other at the speed of sound (when you for instance push a chair, the atoms collide with each other in a chain reaction at the speed of sound, making the chair move).
How can then airplanes fly faster than the speed of sound without something going crazy?
 A: A sample of a gas (air, for example) is composed of molecules which are in random motion, having some mean velocity. The mean velocity of gas molecules in a sample is represented by a quantity we call temperature. The speed of sound in a gas varies with temperature, i.e. the mean velocity of its constituent molecules. The faster the constituent molecules move, the faster a sample can propagate a sound wave.
An object can move through a gas at any velocity; it simply pushes the gas molecules out of the way, accelerating them as much as it takes for them to move out of the way. In a manner of speaking, no object will move faster than the speed of sound in the medium it is traveling because it will collide with, accelerate, and effectively heat up the particles it encounters, increasing the speed of sound immediately in front of the object.
A: The speed of sound in air determines how fast wave phenomena propagate in the air through collisions between molecules, it does not determine how fast other objects immersed in the air travel.
If an object in a fluid like air begins at rest, then the speed of that object after some time is determined by the net external force on that object.  Say, for example, that an airplane's engines exert a force $F$ on the plane, then the net force on the plane will be
\begin{align}
  F - F_\mathrm{air}
\end{align}
where $F_\mathrm{air}$ is the force of the air on the plane, namely air resistance.  As long as this quantity is positive, namely as long is the force of the plane's engines exceeds the resistive force of air, the plane can go faster and faster provided it's made of a material that can withstand the air resistance and heat that will result.
Note, this is a super-simplified model, but it suffices to get the main point across; the speed of sound in the air does not limit the speed of other objects traveling within it, only the speed of sound waves that propagate within it.
A: Atoms interact between each other via various forces, like gravitation or electromagnetic forces. These forces propagate at c (speed of light in void) which is much higher than the speed of sound, hence your assumption "atoms interact with each other at the speed of sound" is wrong.
Otherwise, your description of a movement of a solid (the chair) seems correct but I see no reason why this should limit the speed of the whole chair.
A: 
How can then airplanes fly faster than the speed of sound without something going crazy?

All kinds of things do go crazy.
It's widely reported that Chuck Yeager was the first pilot to break the sound barrier. That's not true. What is true is that Yeager was the first pilot to break the sound barrier and live to tell the tale. A number of other pilots, most notably Geoffrey de Havilland, Jr., crossed the sound barrier prior to Yeager's historic flight but did not live to tell the tale.
De Havilland's plane was subject to extreme oscillations as it approached the speed of sound. De Havilland's head hit the canopy, breaking his neck. Another problem with crossing the sound barrier is changes in control. Various World War II aircraft performed "controlled flight into terrain". Many of these incidents are purported to have resulted from the plane crossing the sound barrier during a dive.
