Do pilots have the "ground speed" information available?
If so, how is the ground speed measured? Or do they rely on GPS info to calculate it?
There are generally two ways of determining the ground speed from within the aircraft.
The first is by using an inertial navigation system. This consists of a number of accelerometers and gyroscopes that measure all accelerations and rotations of the aircraft throughout a flight. By mathematically integrating all measurements this navigation system is able to compute the speed and position of the aircraft at any time. The accuracy will degrade over time but ground speed is usually not far off, otherwise position would be far off very soon.
The second method uses external radio signals. Nowadays this is mostly GPS, but ground speed is also determined by using a number of VOR (VHF Omnidirection Range) and/or DME (Distance Measuring Equipment) stations. These navigation beacons are widely available over most continents. A navigation computer would select the right set of beacons and by analysing the rate of change of the beacon bearings / distances the aircraft's speed can be determined.
Modern navigation systems combine all these methods into a one solution. GPS, inertial navigation and DME measurements are then fused mathematically together to obtain a highly accurate and robust position and ground speed.
Ground speed and ground track are available and are determined by GPS. Comparing two GPS derived locations will yield a velocity vector and ground speed is its magnitude. This is also how winds are derived -- the difference between airspeed and heading compared to groundspeed and ground track.
Note that availability varies by airplane and installed equipment. Any modern airliner will have this info on a primary or secondary display. Smaller general aviation aircraft with an technologically advanced cockpit will have this info as well. Smaller general aviation planes with just a GPS unit will have access to the information but it may not be convenient to reference. Those without even GPS will only be able to derive groundspeed from airspeed and reported winds with a flight computer manually.
A simple definition is:
Ground speed can be determined by the vector sum of the aircraft's true airspeed and the current wind speed and direction; a headwind subtracts from the ground speed, while a tailwind adds to it. Winds at other angles to the heading will have components of either headwind or tailwind as well as a crosswind component.
The issue with using GPS to determine airspeed is that a GPS does not know what the winds are. Although, weather forecast data can provide approximate wind velocity (direction and speed), it is not precise enough for situations where a few knots makes the difference between life and death.