So, you've soldered together your kit, installed your Geiger tube, switched it on, and... nothing happens.
(You should get a click every 3-5 seconds at normal background radiation levels (10-30 CPM).
What to do?
First, make sure the batteries are fresh (or fully charged). Note that some batteries (including the Rocket brand batteries that ship with some versions of the kit) may occasionally get stuck in the battery holder and won't make contact with both terminals. Try pushing each battery away from the spring (negative) side of its compartment so that the positive end makes contact with the metal terminal on the holder. If you have a multimeter, make sure you read about 3V across the battery pack terminals on the back side of the PCB.
If your kit still isn't working, the first step in debugging is to check all of your solder joints. Reflow and add more solder to any joints that look cracked and dull or don't have enough solder on the pad - the solder should make a nice hershey-kiss shape on the pads, with the lead sticking up through the top of the solder joint. The places I usually see bad solder joints the most are the terminals of the battery pack and the piezo speaker.
Double-check that the transistors and diodes are installed in the correct direction.
Make sure FJN3303F is installed in the spot for Q1, and the 2N3904 transistors are installed in Q2 and Q3. If the transistors are in the wrong spot, the kit won't work!
When you turn the kit on, you should hear a faint high pitched whine from the speaker. This whine tells you that the HV supply is working.
If you hear the whine, try rotating VR1 . You should start hearing clicks even with VR1 adjusted at a very low level, but with fresh batteries the correct adjustment is usually about mid-way on the dial. The exact position of VR1 should not be very important. To measure the high voltage, you need to put a 1 GigaOhm resistor in series with your multimeter. An ordinary digital multimeter will give false readings without this resistor. More info here: http://ea4eoz.ure.es/hvprobe.html Measure from TP2 to ground (pin 3 of the PULSE connector is a convenient ground).
If you use a piece of wire (or your fingers) to short across the + and - clips for the tube you should get a click and an LED flash. Try this with the tube installed, and then removed, you should get a click in both cases.
If this doesn't work, try briefly shorting TP1 (the solder pad in the middle of the board labeled GM_PULSE) to ground. One convenient place to get ground is pin 3 of the PULSE connector, this is the pin all the way in the upper right corner of the board, next to the 'E' in PULSE. Or you can touch the wire to the spring terminal on the left side of the bottom battery, that is also ground.