I have made the morse code telegraph machine and need to test the effective range, so I have to work out the resistance and how far apart the bases could be to still work. I also have to test the effective bandwidth. Can anyone help me with how i go about doing this? I would really appreciate any help.

If you used the signal line to switch a transistor operating the light / buzzer, you'd have much less current in the signal line and less voltage-drop.

L

Bandwidth: How fast is you "hand" -- how quickly can you bang out morse code accurately? How quickly can you understand it? For a beginner, those will probably be the limiting factors. For someone more experienced, the fact that this paper key is not very responsive, and that the buzzer takes a moment before it starts sounding, will be the limits -- which is part of why traditional telegraph sounders used a click rather than a buzz, and why the traditional sending key is a more complicated (and precise, and adjustable) mechanism than the simple paper-spring key shown in that Instructable.

So to figure out how much resistance is in the circuit, can I just use the formula V=iR?

And could I use an ammeter to measure the current? How would I need to attach it?

But, yes, you could do it by hand. First measure the voltage across the battery when the circuit is closed (it's probably less than the battery's rated output), then put the ammeter in series with the battery and measure how much current is drawn. Calculate appropriately.

(I believe there are instructables describing the uses of a multimeter. They'll answer this question in greater detail.)