Calibrated Airspeed (CAS)
Calibrated airspeed is the default airspeed type that is selected when you start your aircraft up. Calibrated airspeed is basically the actual airspeed being computed by the air data system (as shown on the analog airspeed indicator), corrected for errors introduced by the air data system. This is determined by a calibration done on a per-aircraft basis. Calibrated airspeeds are given in KIAS (knots indicated airspeed - IAS is the "uncalibrated" airspeed, but there is usually a small enough difference between IAS and CAS that they are interchangeable).
True Airspeed (TAS)
True airspeed is equivalent to CAS corrected for non-standard temperature and pressure. Your airspeed indicator is calibrated based on the standard pressure and temperature at sea level. If the actual pressure and temperature are different from this value (hint: they always are) your calibrated airspeed will be incorrect. This error is negligible at ground level, but as you start to climb, your CAS will begin to diverge more and more from your TAS (generally speaking, your CAS will become lower and lower relative to your TAS as the air density decreases). TAS is calculated from CAS using a simple calculation which takes into account the air's density. True airspeeds are given in KTAS (knots true airspeed).
Ground Speed (GS)
Your ground speed is equivalent to the speed your aircraft is actually travelling over the ground. If you are at 400 KTAS going straight up, your ground speed is 0. In addition, if you are at 400 KTAS but have a 100 knot headwind, your actual ground speed is 300 knots. This is generally computed using the aircraft's INS. Ground speeds are given in plain knots.
The Mach number is a ratio between your aircraft's speed and the speed of sound. It is extremely important to note that there is no one-to-one mapping between airspeed and Mach number - this is a common misconception. The actual speed of Mach 1 depends primarily on the outside air temperature. Mach 1 is approximately 660 knots at sea level on a standard day, but is much slower at high altitudes, especially when the temperature is cold. The Mach number is important because the aircraft's handling characteristics change somewhat when your airspeed is close to Mach 1 (~Mach .8*-1.2, the transonic region). You will notice some turbulence when flying at these speeds.
Your airspeed and Mach number are displayed on the Heads Up Display, as well as the analog airspeed indicator on the main instrument panel. You can select which type of airspeed is displayed on the HUD using this switch on the right console.