Fault finding (Basic)
The key to fault finding electronic circuits is to understand how they are supposed to work. Otherwise you can only use the 'shotgun' approach of testing and replacing individual components until it (hopefully) works.  

The procedure I generally follow is:-


Visual inspection - check for burnt or cracked components, dry joints etc.
Identify components by part number etc. and read the datasheets of devices I am not familiar with.
Trace the circuit and draw a schematic.
Make resistance/voltage/current measurements.
Figure out how the circuit is supposed to work, and compare theory to actual measurements to isolate where it is going wrong.


Sometimes the fault becomes apparent in steps 1 or 2, but when it doesn't then drawing a schematic is essential for all but the simplest circuits. While it may take a little time to do, it saves time in the long run. It can also be combined with the earlier steps. 

When you are not sure what certain components are for, it's time to find out! Think of it as not just fixing a fault, but also gaining general knowledge for the future. 

Your 'transistor' is probably an SCR or TRIAC. You should read and understand its datasheet (if available, or look at similar devices) and the circuits it is typically used in. The 50M&ohm; resistance may just be due to leakage through active components. Most meters use a low voltage to measure resistance, which is not enough to fully turn on diodes etc. At 240V the situation could be very different.   

Mains power is dangerous. Don't attempt a repair until you are sure about what you are doing! Be very careful when testing the device under power. Use an isolating transformer if possible, and clip leads or solder wires onto the PCB rather than touching the meter probes to it. Power the device through a light bulb to limit current in case of a short.