It takes about 12 hours for the Moon to travel from horizon to horizon, so I suggest waiting until Full Moon when the Moon rises just after sunset. Note this position on the eastern horizon. You then have to make a few more observations. Draw a semicircle with the left corner labeled East, the middle labeled South and the right corner labeled West. The upper part of the semicircle is the zenith position right over head. Draw a few landmarks and buildings on this sketch to orient yourself. Then you plot the position of the Moon at various times, say 2 hours apart, across the sky relative to your landmarks. Estimate how high above the horizon they are by 1) either building a simple theodolite from a protractor and a piece of string with a weight on it, or estimate it yourself. A point half way to the zenith position from the horizon is 45 degrees; a point 1/3 the way up is 30 degrees and so on. You do not have to do this all at once, but can do this over the course of several weeks when the Moon gets to more favorable positions during its path across the sky. This path does not change significantly from day to day, so you do not have to follow the Moon continuously over the course of a single evening to trace it out!