It depends a lot on the type of object.
If it's a star, we can observe the red or blueshift of its emission to get an estimate of its velocity towards or away from us, and observe it with telescopes to measure its transverse motion.
If it's an electron, you might need to produce multiple subjects moving the same way and measure different ones at different times realtive to some reference event (say, move a phosphor screen closer or nearer to an aperture the particles are flowing through, to see how they move after they exit the aperture)
If it's an airplane you aren't in, you can track it with a radar system.
If it's a car, there are lots of approaches you might take. You can record the readings from its speedometer, or use a GPS system, use a radar system from outside the car, or just track it with a theodelite, to measure its location.
Depending how you measure the object's location, there are going to be more or less (but never no) errors in your measurement. You'll have to work out how to deal with these.
Then you can fit the measurements to a polynomial or any other function you think is a likely candidate to describe the object's motion. If you're tracking an airplane with radar, this is one of the most studied problems in signal processing, and is the reason the Kalman filter was invented (a technique that would apply well to your vehicle tracking problem also).