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The basic way to measure precession is to make accurate observations of the positions of fixed celestial objects over time. Hipparchus was the first person (at least where it is generally accepted) to have measured precession (in 134 BC). It is common to fix the position of stars relative to the point where the Sun crosses the celestial equator on the Autumnal Equinox (the day when there is exactly 12 hours of sunlight). Hipparchus noticed that the position of stars relative to this point had moved between his measurements and some similar measurements he had records of from 150 years previously. He reasoned that this must mean that the point of the equinox had moved and attributed this to the precession of the sphere of the stars (the Earth was fixed in his day). More modern measurements are exactly equivalent, but we can just measure positions more accurately now.
There are some claims that the Sumerians also measured precession, but I don't believe that they are commonly accepted. It seems to be based on the fact that they counted in multiples of 60 and that the length of the precession cycle is 26,000 years which is about 60*1200*360/1000 years. The 1200 and 360 also have some significance in Sumerian culture - interestingly the Sumerians gave us the degree (360 in a circle - they also had 360 days in their year) and the second/minute (there being 60 of them is no coincidence).