Air is composed of particles in a state of matter called a gas. Gases exist when the atoms or molecules that are present aren't joined together; when they are joined loosely they form a liquid, while a tight packing of molecules creates a solid. You can think of a gas as a bunch of little balls zooming around in random directions.

The Pressure of a gas is determined by how tightly packed together those independent particles get-- if each particle is a long distance from any other particles, that's a very diffuse gas. If we take a piston filled with a diffuse gas and push the piston in-- putting the same amount of gas particles into a smaller volume, then the pressure of the gas increases.

The speed at which the individual particles of the gas move, however,isn't dependent on the pressure-- you could have a diffuse gas of slow or fast moving particles. The speed of the particles is dependent on the temperature. In fact, temperature IS the amount of motion that particles undergo. For example, when you heat a glass of water in the microwave, the microwave energy is making the water molecules move faster, which means the same thing as increasing the temperature.

So, no-- air particles don't move any faster when they are under pressure. That is, unless you puncture a hole-- if you had a container of air under pressure, and punctured a hole, the pressure inside would attempt to normalize with the pressure outside, causing the air to rush out very quickly-- and potentially dangerously.