This is the second electronic circuit I ever made, so it's pretty basic, suitable for beginners.
Step 1: The circuit
I didn't have components to make a high voltage power supply, so I improvised using a transformer from a 10V AC adapter (the little black box you plug into the wall to convert 120VAC to 10VDC). It is not that important to use a 10V transformer, I'm sure anything in the 3V to 12V range would work. Just break open the plastic box and remove the transformer. I used a 1.5V battery (C cell) and a pushbutton switch on the 10V side of the transformer, and the high voltage is generated on the 120V side. You will have to experiment with which polarity to connect the 120V wires of the transformer so as to generate the correct polarity of high voltage, with the wrong polarity the voltage generated will be much smaller.
Tech note: How the high voltage circuit works .
The output voltage of a transformer is proportional to the rate of change of the input current. When you press the pushbutton switch, a fairly large current builds up in the low voltage side of the transformer. When you release the switch, this current instantly drops to zero, and since the rate of change of current is very high (dropping from a large current to zero in a very short time), a high voltage spike is generated on the high voltage side of the transformer. High voltage generating circuits work in much the same way, but they use transistors to switch on and off the current in the low voltage side of the transformer. Since the high voltage is generated only when the button is released, you want to press the button for only a short time so you don't waste the battery current.
The generated high voltage is stored on capacitors and is regulated to 500V using three high voltage Zener diodes in series (200V, 200V, and 100V, adding up to 500V). Such high voltage Zener diodes are not common, you will have to get them at a specialty electronics distributor. Of course whatever reasonably high voltages you can find (for example 1N5271 through 1N5279 are rated 100V through 180V), just add enough in series to equal the voltage rating of the Geiger tube. Make sure to use capacitors rated at least 1000V, and put enough in parallel to add to about 0.02 microfarads. Note that you can't put lower-voltage capacitors in series to get a higher voltage rating, this doesn't work. The Geiger tube I had required 450-500V, other tubes require different voltages, 900V is common, if so use more Zener diodes accordingly.
The diode labeled "1000V PIV" is a rectifier diode with 1000V peak inverse voltage (or peak reverse voltage), to rectify the high voltage (for example 1N4007). I actually used two of these in series because with just one the leakage current was too high, and the high voltage decayed too quickly. If you have lower voltage diodes such as 1N4004 you can put several of them in series. I opened a compact fluorescent light bulb once and it contained several 400V PIV diodes, 3 to 5 of those in series would work. The two unlabeled diodes are 1N914 type.
The signal from the Geiger tube is strong enough to trigger a 555 timer. The 555 timer then drives a small speaker directly, making a 'click' noise each time the Geiger tube detects radiation. Use an IC socket for the 555 timer chip, so you don't fry the chip when trying to solder it into the circuit, also it makes it easier to replace if you make a wrong connection and fry the chip.
Here is a similar design which you can also refer to for other ideas and information: