Assuming I only have a simple telescope, how can I determine the size of the moon, its angular velocity and distance from earth?
Finding the distance is tricky. To find the distance you need to carefully note the position of the moon (relative to the stars ) from two different places, *at the same time *. The moon will appear to be in a slightly different position from the two viewpoints (an effect called paralax). That done, finding the distance, and size is a simple exercise in trigonometry, and Khan academy has a page which shows hoebto find the distance to the moon.
Greater accuracy could be obtained by measuring the time that a star is hidden by the moon, from two locations, and using that to determine the position of the moon to greater accuracy than possible by direct measurements. Hipparchus apparently used a solar eclipse to obtain two positions atbthe same time needed for calculation of paralax.
Knowing the distance makes finding the diameter and speed simple to do, by observing the angular size, and using a little trig. Working with photographs takenby the telescope is convenient but not strictly necessary.