The main thing is that small pressure changes will not race ahead of the airplane once it flies at supersonic speed. Instead of "alerting" the air with a gentle change in pressure, it "surprises" them all of a sudden. This causes rapid pressure changes, called shocks.
The next important effect is that the magnitude of density changes becomes bigger than that of pressure changes in supersonic flow. At very low speed, you will find only pressure changes because the flowing medium behaves like an incompressible gas. At high supersonic speed, however, pressure changes are small and the main flow effects are on the local density of the gas.
In subsonic flow, the oncoming air is first decelerated ahead of the wing, then swiftly accelerated when it flows around the strongly curved nose section of a wing. This acceleration is the consequence of the wing's curvature. See it this way: If the airflow would be along a straight line, it would move away from the surface, creating a local vacuum. In reality, the air settles at a compromise between the straight path and following the contour, creating decreasing pressure along a surface with increasing curvature and increasing pressure along surfaces with decreasing curvature. More precisely, it is always in an equilibrium between inertial, viscous and pressure forces.
This suction not only bends the airflow into following the wing's contour, but also accelerates the air ahead of it. The lower the pressure, the more the air speeds up, such that the total energy of air (the sum of pressure and kinetic energy) stays constant. Therefore, pressure and local speed change in sync.
If the wing moves at high subsonic speed, the curvature-created suction accelerates the flow such that it reaches supersonic speed. Now something odd happens: Supersonic flow accelerates further when subsonic flow would decelerate. This is caused by the change in density which happens at supersonic speed. Now we have a supersonic pocket of air on the upper surface of the wing where speed increases and density decreases downstream, and the surrounding subsonic air sees little change in density.
This cannot last, and at some point this supersonic pocket collapses. This happens instantly in a shock, and in a straight shock density increases suddenly and speed decreases such that the Mach number after the shock is the inverse of the Mach number ahead of the shock. Aft of the shock, you have subsonic flow again.
Once the aircraft accelerates to Mach 1, this shock moves aft to the trailing edge. Now the forward parts of the airplane are fast enough to "surprise" the air they fly into, and the consequence is a straight shock at those parts (mainly the fuselage tip, air intakes and straight leading edges).
If the aircraft accelerates further, the straight shock is bent backwards, because layers of air further away from the fuselage tip or the leading edge will "learn" later of the arrival of the airplane. The result is the Mach cone of a shock, and also the trailing edge shock becomes oblique. Both leading edge and trailing edge shock produce the double-shock system that causes sonic booms, because these shock waves can travel all the way to the ground (if the local speed of sound in the air permits). See the picture above (shamelessly lifted from Wikipedia) which shows how small pressure changes propagate away from the moving aircraft.