How are the velocity and range of a spacecraft measured from ground based stations? For example we can go on NASA's website and look at the real time data for the mission status of their past spacecrafts, like Voyager. How do they obtain this data?
The speed of a spacecraft may be measured from ground using the doppler effect. If the spacecraft sends a signal with a known and very precise frequency, the velocity may be determined by measuring the frequency shift caused by the movement of the spacecraft relative to ground.
Measuring the distance is possible by measuring the delay of a signal from ground to spaceship and back. The known speed of light is used to calculate the distance from the delay time.
See https://en.wikipedia.org/wiki/Doppler_effect and https://en.wikipedia.org/wiki/Radar .
The measurement of distance is done in the same way as using Radar, but when the spacecraft sends a signal back using its transmitter, the method may be used for very long distances too which could not be measured using passive Radar. This way the speed and distance of the Voyager probes may be measured.