Galileo built his first telescopes without any theory of how they worked, with intelligent trial and error, but we have the benefit of a simple theory that is easy to use and easy to picture.
We can start with something we already mentioned in connection with perspective painting. Parallel lines in space can be regarded as intersecting at infinity (the vanishing point in a painting). Turning those lines around, a point at infinity, like a star, emits light that appears to follow parallel lines in our vicinity. When we look at a star, those parallel lines are focused onto our retina and are represented to our consciousness as a bright point. A schematic diagram of the eye looking at starlight is shown below: Click here to activate it. The eye is shown as a red rectangle with a lens on the front. The back of the eye is the retina. If you click the green region at the left, you will see a white dot that you can drag with the mouse to change the direction of the star -- notice how the image point on the retina changes. For any lens, the distance behind the lens at which the image of a star would form is called the focal length. If you click on the lens you will see a white dot at the focal length. By dragging it with the mouse, you can change the eye to one that is farsighted or nearsighted -- the image forms in the wrong place, not on the retina, as it should.
You will have some glass lenses to experiment with. Determine their focal lengths by forming images of distant sources (they need not be stars, of course -- anything far away will do). One of the lenses will have a negative focal length. That means you won't be able to see it form a real image. In the applet representing the eye, drag the focal point THROUGH the lens to the wrong side -- you will then see what a negative focal length lens does to light. Instead of converging to a focus, the starlight diverges, as if it were emanating from the left hand focal point.
Suppose you have two lenses with focal lengths f1 and f2. You can make a telescope simply by placing them a distance f1+f2 apart! Try it, and look through both lenses at some distant objects. Whether you get magnification or demagnification depends which way you turn it, of course. The magnification should be just f1/f2, the RATIO of the focal lengths.
You couldn't determine the focal length of the negative focal length lens before, but now you can -- make a telescope by trial and error, and then figure out the focal length f2 by realizing that the length L of the telescope is f1+f2 and you know both L and f1. Is the magnification f1/f2? You can estimate magnification by keeping both eyes open and comparing the telescopic image with the unmagnified image you see in the other eye.
Galileo was very interested in how much his telescopes magnified, because, as always, he was trying to make QUANTITATIVE measurements with them. He realized, as artists before him had realized, that the appropriate measure of how big objects appear to us is ANGLE. The angular size of something of width w at a distance D is w/D. If D gets bigger, the object looks smaller, because D is in the denominator. That is the familiar rule for making things smaller in a painting if you want to represent them as farther away. Measure the angular size of the field of view of one of your telescopes by measuring both w and D for some convenient object that just fills the field of view (at some distance away -- not too close). The full field of view then has angular size w/D. Now you can measure the angular sizes of anything you see in the telescope just by estimating what fraction of the full width the object occupies. Galileo used this method. Choose a distant object and measure its angular size.In case you are interested, there is an applet of an eye looking at a star through a telescope at the bottom of this page. Activate it here, and feel free to alter the components by dragging them around.